Intuitionism(en.wikipedia.org)
en.wikipedia.org
Intuitionism
https://en.wikipedia.org/wiki/Intuitionism
178 comments
To paraphrase Wittgenstein, is it more likely that we “discovered” chess or that we invented it? He suggests all of mathematics should be thought of in this way, as more and more complex ways of stating tautologies.
> fools us into thinking there is some inherent order -- which there isn't.
Bold claim :)
Even while maintaining a willful agnosticism about Platonic realism, it seems clear that the business of doing mathematics - intuitionistic or otherwise - depends on the ability to state and follow unambiguous rules, or else how to establish a proof within some axiomatic system or other?
But if mathematicians have this ability, even as an asymptotic ideal, and are able to make judgments as to when rules are not being followed correctly, mustn't this in turn depend on some pre-existing regularity (i.e. order) in the universe? Ineluctable physical law seems a natural candidate for this order.
Bold claim :)
Even while maintaining a willful agnosticism about Platonic realism, it seems clear that the business of doing mathematics - intuitionistic or otherwise - depends on the ability to state and follow unambiguous rules, or else how to establish a proof within some axiomatic system or other?
But if mathematicians have this ability, even as an asymptotic ideal, and are able to make judgments as to when rules are not being followed correctly, mustn't this in turn depend on some pre-existing regularity (i.e. order) in the universe? Ineluctable physical law seems a natural candidate for this order.
In other words, the Cartesian principle extended to two parties?
Cogito sicut tu cogitas, ergo sum sicut structus es: I think as you think (at least for a moment); therefore, I am, at some level, structured as you are (at least for a moment); and therefore, there is order in the universe, at least in the temporary alignment of our structures that enables us to think along these same lines?
(Saying nothing, of course, of whether you "exist." I may be a brain in a vat, and your thoughts may be the emission of an evil demon running a simulation — but that still means that my thoughts and the evil demon's simulation share structure that implies an underlying shared set of computational axioms!)
Cogito sicut tu cogitas, ergo sum sicut structus es: I think as you think (at least for a moment); therefore, I am, at some level, structured as you are (at least for a moment); and therefore, there is order in the universe, at least in the temporary alignment of our structures that enables us to think along these same lines?
(Saying nothing, of course, of whether you "exist." I may be a brain in a vat, and your thoughts may be the emission of an evil demon running a simulation — but that still means that my thoughts and the evil demon's simulation share structure that implies an underlying shared set of computational axioms!)
Yes, thank you!
Thank you for your reply. I'll try some more! :)
I noticed that in nature (i.e. in the physical universe) there appears to be no logic. There are not even two things exactly the same, as far as I can tell. So that lead me to believe that "counting" (or abstraction) is not even a property of the universe, but possibly only a human (or animal, or Turing machine) construct. To me, logic only starts to occur when a very complex amalgam of matter comes together [1] to realize a discrete switch. Using these discrete switches, organisms build memory and abstraction mechanisms, start counting, and do mathematics.
So my thesis is that things that are considered to be "basic" by most, such as logic or mathematics, are in fact quite specific systems built on top of chaos. From there, it remains to be proven that all physical laws that we observe are no more than projections of the chaos onto such a system.
Perhaps a metaphor that helps to take on my perspective, is to look at a screen filled with random noise, and then observe some patterns in there. Now replace the screen with an infinite dimensional set of chaos, and then observe a pattern that is our universe. With the added twist that we are part of this chaos, and observing the pattern, possibly in the form of physical laws.
Of course, there are many problems with this theory. What does the chaos reside in? How can there be discernible parts in the chaos? Is time an emergent property inside the chaos? How can it be that the patterns that we observe are so consistent?
However, to me this theory seems more fruitful than merely accepting that we cannot say anything about things that science cannot observe, or that some deity created all this.
[1] With "comes together" I do not refer to a dynamic process, but to the accidental occurrence of stuff in such a shape or form. Obviously, my ridiculous theory asserts a chaos chock full of dimensions, where time and space are but supporting actors.
I noticed that in nature (i.e. in the physical universe) there appears to be no logic. There are not even two things exactly the same, as far as I can tell. So that lead me to believe that "counting" (or abstraction) is not even a property of the universe, but possibly only a human (or animal, or Turing machine) construct. To me, logic only starts to occur when a very complex amalgam of matter comes together [1] to realize a discrete switch. Using these discrete switches, organisms build memory and abstraction mechanisms, start counting, and do mathematics.
So my thesis is that things that are considered to be "basic" by most, such as logic or mathematics, are in fact quite specific systems built on top of chaos. From there, it remains to be proven that all physical laws that we observe are no more than projections of the chaos onto such a system.
Perhaps a metaphor that helps to take on my perspective, is to look at a screen filled with random noise, and then observe some patterns in there. Now replace the screen with an infinite dimensional set of chaos, and then observe a pattern that is our universe. With the added twist that we are part of this chaos, and observing the pattern, possibly in the form of physical laws.
Of course, there are many problems with this theory. What does the chaos reside in? How can there be discernible parts in the chaos? Is time an emergent property inside the chaos? How can it be that the patterns that we observe are so consistent?
However, to me this theory seems more fruitful than merely accepting that we cannot say anything about things that science cannot observe, or that some deity created all this.
[1] With "comes together" I do not refer to a dynamic process, but to the accidental occurrence of stuff in such a shape or form. Obviously, my ridiculous theory asserts a chaos chock full of dimensions, where time and space are but supporting actors.
> built on top of chaos.
I don't think you've taken on the full force of the argument that regularity in human activity (such as building systems) requires a source of order for it not to simply dissolve into chaos itself.
> How can it be that the patterns that we observe are so consistent?
How can we claim to discern consistency (or inconsistency) without the ability to follow a rule correctly? And how can we follow a rule without a source of order or regularity in the cosmos? Wouldn't it be like trying to build a the Eiffel tower out of live slugs?
If you insist on an absence of order in the physical universe, the onus is on you to explain how regularity in human activity (required for mathematics of any kind) can be achieved without it.
This is not an argument for Platonic realism, BTW, or against intuitionism, roughly construed as the view that "mathematics is a creation of the mind" as per [1], or in your formulation that 'mathematical abstractions are not part of the physical universe' (if I understand what you're saying). You can perfectly well believe that mathematics is a mental construct and at the same time acknowledge that it's possible to observe regularities and order in the cosmos. If you want to insist that the regularity doesn't come from physical law, then I find it hard to see how you'll escape from some kind of Platonic belief in a non-physical realm that serves as the source of order :)
In your TV screen dots analogy, isn't it usually thought that patterns appear only because of the structured, generative activity of law-observing physical components, specifically the neurons comprising your grey matter?
1: https://plato.stanford.edu/entries/intuitionism/
I don't think you've taken on the full force of the argument that regularity in human activity (such as building systems) requires a source of order for it not to simply dissolve into chaos itself.
> How can it be that the patterns that we observe are so consistent?
How can we claim to discern consistency (or inconsistency) without the ability to follow a rule correctly? And how can we follow a rule without a source of order or regularity in the cosmos? Wouldn't it be like trying to build a the Eiffel tower out of live slugs?
If you insist on an absence of order in the physical universe, the onus is on you to explain how regularity in human activity (required for mathematics of any kind) can be achieved without it.
This is not an argument for Platonic realism, BTW, or against intuitionism, roughly construed as the view that "mathematics is a creation of the mind" as per [1], or in your formulation that 'mathematical abstractions are not part of the physical universe' (if I understand what you're saying). You can perfectly well believe that mathematics is a mental construct and at the same time acknowledge that it's possible to observe regularities and order in the cosmos. If you want to insist that the regularity doesn't come from physical law, then I find it hard to see how you'll escape from some kind of Platonic belief in a non-physical realm that serves as the source of order :)
In your TV screen dots analogy, isn't it usually thought that patterns appear only because of the structured, generative activity of law-observing physical components, specifically the neurons comprising your grey matter?
1: https://plato.stanford.edu/entries/intuitionism/
This argument completely ignores the observer which is bound by the same limits of our processing - indeed they are paired together.
Entering purely theoretical space here: On the timescale of eternity this might be a local pocket of some logical organization but there is no fundamental logic governing everything. Our observation is limited so we can’t perceive chaos, instead evolved to only recognize patterns. Over infinity, pure chaos does not preclude long pockets of what looks like order. What we consider fundamental rules could very well be local phenomenon, which we are a product of.
Of course this purely theoretical - all I’m saying is that intuitism could be true while also math being useful to predict things right now. We could also only exist for an instant and all our memories just construct, but that’s not very useful. It’s more useful to believe in scientific method because what’s repeatable is provable, whereas chaos is by its nature unprovable - which doesn’t make it impossible.
Entering purely theoretical space here: On the timescale of eternity this might be a local pocket of some logical organization but there is no fundamental logic governing everything. Our observation is limited so we can’t perceive chaos, instead evolved to only recognize patterns. Over infinity, pure chaos does not preclude long pockets of what looks like order. What we consider fundamental rules could very well be local phenomenon, which we are a product of.
Of course this purely theoretical - all I’m saying is that intuitism could be true while also math being useful to predict things right now. We could also only exist for an instant and all our memories just construct, but that’s not very useful. It’s more useful to believe in scientific method because what’s repeatable is provable, whereas chaos is by its nature unprovable - which doesn’t make it impossible.
Well, not a possibility I considered, but nothing in the argument depends on the regularity being a permanent feature of the cosmos, just that systematic human endeavours, such as mathematics or indeed meaningful debate, depend on it, so when it goes they go. In that sense, if you wish to consider this conversation meaningful you are kind of ceding the point that for now chaos doesn't, in fact, reign. If you don't consider it meaningful, then why are you having it? :)
Not arguing anything - it's more an interesting thought experiment. This view doesn't change much other than never finding the "true unified theory of everything", which I'm not sure how many people think is truly possible (at least anytime soon) anyway.
It is useful to focus on repeating things, and useless to focus on randomness. But I don't think it's necessarily true that randomness (probably a better word than chaotic since chaotic systems are complex mathematical interactions) doesn't reign. We evolved to take advantage of repeatable things, our sense organs and perception are all focused on things that are repeatable. Our definition of usefulness (what is useful/what isn't) depends on repeatable things. I believe there's a pretty high likelihood that we are blind to anything outside of that, such as true randomness. IE, we literally cannot conceive of true randomness since we are products of an environment that rewarded it.
To your point, I guess this isn't exactly Intuitionism, since Intuitionism says it's a totally human construct, and mathematics has provenly predicted things in nature from purely theoretical models, whereas I just find the part that supposes mathematics isn't a fundamental part of objective reality possible.
Either way, I don't think it changes anything about how we do math or science or anything - how could you even study this? By definition understanding and using things depends on repeatability. If there truly were cracks in it, they by definition couldn't be repeatable. It certainly won't help us get food.
EDIT > If you don't consider it meaningful, then why are you having it?
I just find it interesting since I've had this thought before and seeing what other people think of it.
It is useful to focus on repeating things, and useless to focus on randomness. But I don't think it's necessarily true that randomness (probably a better word than chaotic since chaotic systems are complex mathematical interactions) doesn't reign. We evolved to take advantage of repeatable things, our sense organs and perception are all focused on things that are repeatable. Our definition of usefulness (what is useful/what isn't) depends on repeatable things. I believe there's a pretty high likelihood that we are blind to anything outside of that, such as true randomness. IE, we literally cannot conceive of true randomness since we are products of an environment that rewarded it.
To your point, I guess this isn't exactly Intuitionism, since Intuitionism says it's a totally human construct, and mathematics has provenly predicted things in nature from purely theoretical models, whereas I just find the part that supposes mathematics isn't a fundamental part of objective reality possible.
Either way, I don't think it changes anything about how we do math or science or anything - how could you even study this? By definition understanding and using things depends on repeatability. If there truly were cracks in it, they by definition couldn't be repeatable. It certainly won't help us get food.
EDIT > If you don't consider it meaningful, then why are you having it?
I just find it interesting since I've had this thought before and seeing what other people think of it.
Something to consider in the context of "repetition", is that it requires abstraction, and possibly memory. As noted before, I do not see any kind of repetition (identical things, counting) in nature. I think abstraction and memory are both emergent properties from human brains (or machines, brains in other mammals, octopuses, etc.) My pet theory also initially discards "things", because that again requires abstraction.
For reference, my views are somewhat related to "emergentism", "connectionism", and "realism", but I haven't found a school of philosophy that I feel comfortable with.
> how could you even study this?
This is indeed the biggest challenge. I am currently studying this from a conceptual art perspective, because philosophy and science do not seem adequately equipped for this kind of problem.
For reference, my views are somewhat related to "emergentism", "connectionism", and "realism", but I haven't found a school of philosophy that I feel comfortable with.
> how could you even study this?
This is indeed the biggest challenge. I am currently studying this from a conceptual art perspective, because philosophy and science do not seem adequately equipped for this kind of problem.
Thanks again for taking the time for a thoughtful reply. I am aware that I'm using terminology very loosely, and I omit many details that may be required for a full understanding.
With respect to the full force of the argument: I assume that the "regularity" stems from the physical systems that make up our brain. Just as replication through DNA offers some stability in life, the shape of our neurons (and perhaps the dynamics of space-time, and the laws of physics) offer some regularity in the chaotic universe.
In my view, this regularity is but an accidental blip in the totality of existence, but to us, who cannot observe the rest of the chaos, it seems fundamental -- which from the universal context, it isn't.
The biggest problem that I cannot get around is that I somehow assume this chaos exists, and allows for things to exist inside of it. I do not know how to provide arguments for that, other than the negative one that it seems highly unlikely that "there is something rather than nothing". Likeliness, and the fact that I can define these abstract concepts, only make sense in the realm of human thought, so I am sort of stuck in a recursive loop there.
With regards to the second part of your reply, again it is us humans who do the discerning. It is an emergent property of our brain (or possibly of slightly simpler, but still rather complex "discrete switches") that we can discern things. In the underlying universe of total chaos, there is no context, no logic, no measure to discern things.
So, the source of order does arise through physical constructs, that happen to have a certain structure that allows observation. It is humans, mammals, octopuses, computers, that can use this universal form of observation to process input, and then do observation as we know it. So I suppose my idea is some kind of realism, but my reality is nothing more than pure and utter, unbounded chaos. And we live in some corner of that.
The grand claim is that mathematics is nothing more than the result of some self-observing shapes in the chaos that is existence.
Again, I feel sorry for all the readers who try to make sense of all my overloaded concepts. I wish I had the skills to write down my thoughts more rigorously. Or perhaps someone can save me a lot of time [1].
[1] https://xkcd.com/386/
With respect to the full force of the argument: I assume that the "regularity" stems from the physical systems that make up our brain. Just as replication through DNA offers some stability in life, the shape of our neurons (and perhaps the dynamics of space-time, and the laws of physics) offer some regularity in the chaotic universe.
In my view, this regularity is but an accidental blip in the totality of existence, but to us, who cannot observe the rest of the chaos, it seems fundamental -- which from the universal context, it isn't.
The biggest problem that I cannot get around is that I somehow assume this chaos exists, and allows for things to exist inside of it. I do not know how to provide arguments for that, other than the negative one that it seems highly unlikely that "there is something rather than nothing". Likeliness, and the fact that I can define these abstract concepts, only make sense in the realm of human thought, so I am sort of stuck in a recursive loop there.
With regards to the second part of your reply, again it is us humans who do the discerning. It is an emergent property of our brain (or possibly of slightly simpler, but still rather complex "discrete switches") that we can discern things. In the underlying universe of total chaos, there is no context, no logic, no measure to discern things.
So, the source of order does arise through physical constructs, that happen to have a certain structure that allows observation. It is humans, mammals, octopuses, computers, that can use this universal form of observation to process input, and then do observation as we know it. So I suppose my idea is some kind of realism, but my reality is nothing more than pure and utter, unbounded chaos. And we live in some corner of that.
The grand claim is that mathematics is nothing more than the result of some self-observing shapes in the chaos that is existence.
Again, I feel sorry for all the readers who try to make sense of all my overloaded concepts. I wish I had the skills to write down my thoughts more rigorously. Or perhaps someone can save me a lot of time [1].
[1] https://xkcd.com/386/
>There are not even two things exactly the same, as far as I can tell.
https://medium.com/physics-as-a-foreign-language/how-do-we-k...
https://www.popularmechanics.com/science/news/a27731/what-if...
https://medium.com/physics-as-a-foreign-language/how-do-we-k...
https://www.popularmechanics.com/science/news/a27731/what-if...
Your url-based reply is quite ironic.
The first link is an article that explains that all electrons are exactly identical, suggesting that there are indeed things in the universe that are exactly the same. However, the second link discusses the one-electron idea by Wheeler, which suggests the exact opposite :)
The first link is an article that explains that all electrons are exactly identical, suggesting that there are indeed things in the universe that are exactly the same. However, the second link discusses the one-electron idea by Wheeler, which suggests the exact opposite :)
> There are not even two things exactly the same, as far as I can tell
Aren’t all basic particles defined by the fact that they are exactly the same? And countable things rely on some difference, such as different spatial locations — or else they wouldn’t be countable, they’d just be the same.
While two neutron stars are distinguishable they are also classifiable as a real type of star in a manner that seems to go beyond human perceptual idiosyncrasy.
But I'm a deep Platonist/Pythagorean — so my bias is that "all is number" and the world is made of math. Math is real :)
Aren’t all basic particles defined by the fact that they are exactly the same? And countable things rely on some difference, such as different spatial locations — or else they wouldn’t be countable, they’d just be the same.
While two neutron stars are distinguishable they are also classifiable as a real type of star in a manner that seems to go beyond human perceptual idiosyncrasy.
But I'm a deep Platonist/Pythagorean — so my bias is that "all is number" and the world is made of math. Math is real :)
If the world is made of math, do new physical objects pop into being whenever a mathematician writes down or thinks of a structure or a proof? Or does only some math get to become physical?
In general, your view seems like a definitional issue to me. If you want to call what the world is made of “math”, then what you and I mean by math are two different things, and using the same word to describe them only leads to confusion.
In general, your view seems like a definitional issue to me. If you want to call what the world is made of “math”, then what you and I mean by math are two different things, and using the same word to describe them only leads to confusion.
Well, we try to discover math. That’s straight forward?
Again, definitional issues are critical here.
I can make up all sorts of math that isn't really "discovered", it's more "invented". Most of it won't correspond usefully to the physical universe. See e.g.: https://plato.stanford.edu/entries/formalism-mathematics/ :
> "mathematics is not a body of propositions representing an abstract sector of reality but is much more akin to a game, bringing with it no more commitment to an ontology of objects or properties than ludo or chess."
I take the brevity and lack of commitment to a position in your reply as an indication that you're not really willing or able to defend your position. That's fine, it's a complex topic, as the many articles on the SEP linked above attest. But if you want to claim "the world is made of math", then the onus is on you to define what you mean by that. To me, it looks a little incoherent.
I can make up all sorts of math that isn't really "discovered", it's more "invented". Most of it won't correspond usefully to the physical universe. See e.g.: https://plato.stanford.edu/entries/formalism-mathematics/ :
> "mathematics is not a body of propositions representing an abstract sector of reality but is much more akin to a game, bringing with it no more commitment to an ontology of objects or properties than ludo or chess."
I take the brevity and lack of commitment to a position in your reply as an indication that you're not really willing or able to defend your position. That's fine, it's a complex topic, as the many articles on the SEP linked above attest. But if you want to claim "the world is made of math", then the onus is on you to define what you mean by that. To me, it looks a little incoherent.
Take the brevity as a lack of hubris. The notion that “the world is made of math” is one of the oldest and most influential ideas of all time. If you find Pythagoras, Plato and Newton a little incoherent, that’s not unusual. But the onus doesn’t lie with them (or me). In any case, I remain interested in your ideas!
None of Pythagoras, Plato, or Newton claimed that the “world is made of math”. Also, Aristotle’s philosophy of mathematics is considered an alternative to Plato’s, so trying to seek solace in both at the same time seems inconsistent.
Plato described math as a realm distinct from both the physical world and the world of consciousness. This doesn’t support the idea the “the world is made of math”.
Newton described the world as operating in accordance with the rules of math, but that’s not the same as being “made of math.” Plato’s view is compatible with this: what Frege described as the “third realm”, the realm of abstract objects, can have a relationship with the physical realm without requiring that the latter be “made of” the former.
Aristotle explicitly distinguished between physics and mathematics, saying in his Metaphysics that physics is concerned with things that change, whereas mathematics encompasses things that are eternal, do not change, and are not substances. So Aristotle seems to explicitly reject your view.
As such, I don’t accept your claim that your position is “one of the oldest and most influential ideas of all time.”
> But the onus doesn’t lie with them (or me).
If you make a claim, the onus certainly lies on you to support that claim.
Plato described math as a realm distinct from both the physical world and the world of consciousness. This doesn’t support the idea the “the world is made of math”.
Newton described the world as operating in accordance with the rules of math, but that’s not the same as being “made of math.” Plato’s view is compatible with this: what Frege described as the “third realm”, the realm of abstract objects, can have a relationship with the physical realm without requiring that the latter be “made of” the former.
Aristotle explicitly distinguished between physics and mathematics, saying in his Metaphysics that physics is concerned with things that change, whereas mathematics encompasses things that are eternal, do not change, and are not substances. So Aristotle seems to explicitly reject your view.
As such, I don’t accept your claim that your position is “one of the oldest and most influential ideas of all time.”
> But the onus doesn’t lie with them (or me).
If you make a claim, the onus certainly lies on you to support that claim.
Pythagoras said “all is number.” So, where some claimed that “fire” was the primary constituent of all things and others “earth,” Pythagoreanism held that numbers were the underlying principle. Do you accept this claim?
Newton was a Pythagorean. He even attributed the inverse square law to Pythagoras.
Plato is always hard to pin down, but he describes the immaterial world as crafted by number, prior to the material world.
Newton was a Pythagorean. He even attributed the inverse square law to Pythagoras.
Plato is always hard to pin down, but he describes the immaterial world as crafted by number, prior to the material world.
There are not even two things exactly the same, as far as I can tell.
That is a common definition of identity, two things are the same if all their properties are the same. So by definition there can not be two different things with all their properties the same as this would make them indistinguishable and therefore the same thing. But if you relax this a bit, then for example elementary particles like electrons are - as far as we know - all completely identical up to their position, momentum and spin.
That is a common definition of identity, two things are the same if all their properties are the same. So by definition there can not be two different things with all their properties the same as this would make them indistinguishable and therefore the same thing. But if you relax this a bit, then for example elementary particles like electrons are - as far as we know - all completely identical up to their position, momentum and spin.
I've seen statistics proposed as the force that makes reality, that would be fundamentally random, coherent. But statistics laws are themselves very strong when numbers get big.
> mustn't this in turn depend on some pre-existing regularity
Consider the set of all things, including the ordered and disordered. Consider the operation of taking subsets of that set. Consider that conscious entities such as us only arise in ordered subsets (for some definition of ordered).
Those conscious entities would see their proximal environment as ordered. They might assume that the only things in the set of all things are those things that are ordered in the same way as the proximal environment from which they arose.
Our universe may be just a subset of a subset. That is, our universe might be one kind of universe that has some apparent quantum-field or relativity-geometry based regularity. There might be some number of universes that share that kind of ordering and perhaps not in all of them conscious entities arise.
It is still an interesting question to ask: why would conscious entities arise within this particular kind of ordered universe? But it no longer remains the question of whether or not regularity is a fundamental property of all possible states of existence. And in fact it leaves open the question as to whether or not some kind of consciousness (perhaps very different from our own) could arise in what would appear to us as chaotic universes.
Consider the set of all things, including the ordered and disordered. Consider the operation of taking subsets of that set. Consider that conscious entities such as us only arise in ordered subsets (for some definition of ordered).
Those conscious entities would see their proximal environment as ordered. They might assume that the only things in the set of all things are those things that are ordered in the same way as the proximal environment from which they arose.
Our universe may be just a subset of a subset. That is, our universe might be one kind of universe that has some apparent quantum-field or relativity-geometry based regularity. There might be some number of universes that share that kind of ordering and perhaps not in all of them conscious entities arise.
It is still an interesting question to ask: why would conscious entities arise within this particular kind of ordered universe? But it no longer remains the question of whether or not regularity is a fundamental property of all possible states of existence. And in fact it leaves open the question as to whether or not some kind of consciousness (perhaps very different from our own) could arise in what would appear to us as chaotic universes.
I spent quite some time thinking along these lines, but then I realized that it is very unlikely to be the case in our universe.
If our world would be an "accidental" subset of the universe, with ordering, then how can it be that this ordering is so consistent? Would it not be more likely that we'd live in a world that has only some order, or varying order depending on one's position in space and time? For example, I would expect a lot more miracles to happen, but every physics experiment turns out to be extremely consistent.
This led me to believe that there is another process at play. My thought experiment now assumes that observation enforces a certain kind of consistency in the laws of physics. That is, the systems that we use to observe something, must by their very existence result in very consistent patterns in the chaos. What this precisely looks like and how it operates is left as an exercise to the reader.
If our world would be an "accidental" subset of the universe, with ordering, then how can it be that this ordering is so consistent? Would it not be more likely that we'd live in a world that has only some order, or varying order depending on one's position in space and time? For example, I would expect a lot more miracles to happen, but every physics experiment turns out to be extremely consistent.
This led me to believe that there is another process at play. My thought experiment now assumes that observation enforces a certain kind of consistency in the laws of physics. That is, the systems that we use to observe something, must by their very existence result in very consistent patterns in the chaos. What this precisely looks like and how it operates is left as an exercise to the reader.
> observation enforces a certain kind of consistency in the laws of physics.
That sounds like a typical chicken-and-egg problem.
> how can it be that this ordering is so consistent?
One thing that I didn't explicitly state is that the universe only needs to seem ordered. You are making an assumption that isn't necessarily founded. 99.9% of the time we aren't checking on the ordered state of the universe. It is the exceedingly rare case that we measure with sufficient granularity to expect to see quantum effect, for example.
> I would expect a lot more miracles to happen
I totally understand this. You might expect a table upon which your keyboard rests to change color as we pass through a chaotic portion of some multi-verse. But that assumes order can't exist at the time-scales of universes.
Imagine it like a picture. The set of all possible pictures at 640x480 is massive but finite. The vast majority of images in that space are like white noise. But you've probably seen millions of images at that resolution that are totally coherent. You don't expect the middle block of some specific image to randomly be a different color. Same with a movie. The set of all sequences of 640x480 images at 30 fps with 1 minute duration is finite. It is full of total chaos. But when you watch a movie you don't expect chaos to ensue at any moment.
If you consider our entire universe like a single image from the set of all images or a movie from the set of all movies, it isn't surprising that it is entirely ordered from beginning to end. It just feels weird because the scale in both time and space is so large in the case of the universe. But given the scale of the entire set, such a large subset having internal consistency isn't totally unreasonable.
That sounds like a typical chicken-and-egg problem.
> how can it be that this ordering is so consistent?
One thing that I didn't explicitly state is that the universe only needs to seem ordered. You are making an assumption that isn't necessarily founded. 99.9% of the time we aren't checking on the ordered state of the universe. It is the exceedingly rare case that we measure with sufficient granularity to expect to see quantum effect, for example.
> I would expect a lot more miracles to happen
I totally understand this. You might expect a table upon which your keyboard rests to change color as we pass through a chaotic portion of some multi-verse. But that assumes order can't exist at the time-scales of universes.
Imagine it like a picture. The set of all possible pictures at 640x480 is massive but finite. The vast majority of images in that space are like white noise. But you've probably seen millions of images at that resolution that are totally coherent. You don't expect the middle block of some specific image to randomly be a different color. Same with a movie. The set of all sequences of 640x480 images at 30 fps with 1 minute duration is finite. It is full of total chaos. But when you watch a movie you don't expect chaos to ensue at any moment.
If you consider our entire universe like a single image from the set of all images or a movie from the set of all movies, it isn't surprising that it is entirely ordered from beginning to end. It just feels weird because the scale in both time and space is so large in the case of the universe. But given the scale of the entire set, such a large subset having internal consistency isn't totally unreasonable.
I don’t have anything to add, but I just genuinely love your analogy. Totally helped me see the idea in a different way
> mustn't this in turn depend on some pre-existing regularity (i.e. order) in the universe?
Yes. And such order exists within the observer, the mediators. We are measuring instruments. Scales are bound to the ruler and not to what’s being ruled. So mathematics represents order only in as much as there are people to validate it. Should all rulers be broken and forgotten, then there’s no measure at all. This is the same for those orders not yet established, that is, the future of science. Nevertheless, I do believe there’s a principle which allows order, but it’s not order itself, but a foundation of order, which I believe to be Unit. Unit is not mere duality, because duality implies two, Unit would be more like Cause-Effect, wherein one is the same as the other (either Cause-Cause, or Effect-Effect, doesn’t really matter). This is also different than yin-yang, since each is discrete, and discreetness itself cannot exist prior to Unit. I like to think that all numbers are qualities of Unit, and the whole of mathematical theories are different theories of Unit, so they will be consistent every time Unit is maintained consistently, when something “follows” from what has already “followed”, following some definition of “following”, whatever it may be. It would explain the effectiveness of mathematics in the sense that the whole Universe is Unit out of self-similar Unit. The fact that all of information can be encoded in 0s and 1s is a great illustration of the power of Unit, and the fact that binary streams only makes “sense” upon interpretation is a great illustration of consciousness, which is an expression of Bias. Even if the Universe would change so all of the physical “laws” would mutate, Unit would still persist unchanged. New things would still “follow”. I haven’t come across anyone realizing that a theory of “everything” can exist but be useless, just like the concept of “everything” is useless as a particularity, and theories aim to be particularly applicable, so a theory which really applies to every thing applies to no thing. This would be the utmost conclusion of Gödel‘s incompleteness. How would a theory of absolutely everything be different than an infinitely long ruler without any subdivisions, or even with infinitely many zero-spaced subdivisions? One wouldn’t be able to measure any particular thing with such universal ruler.
Yes. And such order exists within the observer, the mediators. We are measuring instruments. Scales are bound to the ruler and not to what’s being ruled. So mathematics represents order only in as much as there are people to validate it. Should all rulers be broken and forgotten, then there’s no measure at all. This is the same for those orders not yet established, that is, the future of science. Nevertheless, I do believe there’s a principle which allows order, but it’s not order itself, but a foundation of order, which I believe to be Unit. Unit is not mere duality, because duality implies two, Unit would be more like Cause-Effect, wherein one is the same as the other (either Cause-Cause, or Effect-Effect, doesn’t really matter). This is also different than yin-yang, since each is discrete, and discreetness itself cannot exist prior to Unit. I like to think that all numbers are qualities of Unit, and the whole of mathematical theories are different theories of Unit, so they will be consistent every time Unit is maintained consistently, when something “follows” from what has already “followed”, following some definition of “following”, whatever it may be. It would explain the effectiveness of mathematics in the sense that the whole Universe is Unit out of self-similar Unit. The fact that all of information can be encoded in 0s and 1s is a great illustration of the power of Unit, and the fact that binary streams only makes “sense” upon interpretation is a great illustration of consciousness, which is an expression of Bias. Even if the Universe would change so all of the physical “laws” would mutate, Unit would still persist unchanged. New things would still “follow”. I haven’t come across anyone realizing that a theory of “everything” can exist but be useless, just like the concept of “everything” is useless as a particularity, and theories aim to be particularly applicable, so a theory which really applies to every thing applies to no thing. This would be the utmost conclusion of Gödel‘s incompleteness. How would a theory of absolutely everything be different than an infinitely long ruler without any subdivisions, or even with infinitely many zero-spaced subdivisions? One wouldn’t be able to measure any particular thing with such universal ruler.
Does your "principle which allows order" presume the existence of space and time, and more specifically the ordering of time? I would say that in a universe without time, "cause and effect" have little meaning.
I think I lost your train of thought when you say that numbers are qualities of Unit. Does your universe involve only "Unit", or are there other principles at play as well?
I think I lost your train of thought when you say that numbers are qualities of Unit. Does your universe involve only "Unit", or are there other principles at play as well?
The Peano arithmetic hints at the proposition that numbers are gradations within a unified principle, in this case the principle of “succession”. If you start with nothing (zero) and recursively apply the same quality (succession) you get all integers. So we can rationally assume that all numbers are different qualifications of some primitive. As for time and space, they do not fundamentally exist and they do not configure a necessity within the universe. It takes a being able to record and internally persist events for time to appear. Space is similar, being also a referentiality. In essence, time and space are different framings of the same phenomenon. The length distance between two points is also a temporal ratio between the points. It is just two perceptions, two expediences, not grounded in singular reality. I believe “beings” are Unit juxtaposed over Unit, in the spirit of Wheeler ideas [1]. I don’t think there are fundamental necessities other than Unit. Unit is not causality, I have just used the terms for illustration. If there are other fundamental necessities other than Unit then one would need to explain how they came into being, and it would eventually recurse into Unit. From nucleosynthesis to procreation, it’s Unit all over. The way I grasp it, mathematics is a coloring of Unit, just like for seeing wind one need to sprinkle something over it. It boils down to the ascription of Parts within Wholes. Parts are mere subjections.
[1] https://www.reddit.com/r/holofractal/comments/6wnekw/this_be...
[1] https://www.reddit.com/r/holofractal/comments/6wnekw/this_be...
The physical world is a persistent system that exhibits highly consistent behaviour. Because the behaviour is consistent we can describe it in a highly consistent formal language, mathematics.
What Plato called forms are just descriptions. We have a description of what a circle is, and anything that matches that description is a circle.
What Plato called forms are just descriptions. We have a description of what a circle is, and anything that matches that description is a circle.
Highly consistent!? Not really. Some things are, but like OP just said, those are the ones we glom onto. But don't mistake some things for every things, there's a whole big wide world out there. We can hardly describe a ripple in a stream let alone why I've had the 5th argument in five weeks about the order I have to fix the kitchen with me wife, yet my intuition told me it was a comin.
So you consistently perceive streams, the world, have a body, a life, a wife, a kitchen with a flaw that has persisted over time, an order in to fix it. Also the world is consistent enough that you could predict that argument in advance. That sounds like an awful lot of consistency :)
Yes, I've already conceded that some things are consistent. Will you concede that we tend to glom onto them or will I interpret your reply where you once again glommed onto them as a concession? ;)
It's not glomming on when these things are basically the core aspects of reality.
You haven't really come to grips with what a world that was mostly inconsistent would look like.
You haven't really come to grips with what a world that was mostly inconsistent would look like.
> what a world that was mostly inconsistent would look like
Probably one that had trouble evolving high-order lifeforms, aka anthropic principle.
Probably one that had trouble evolving high-order lifeforms, aka anthropic principle.
> The physical world is a persistent system that exhibits highly consistent behaviour.
Thats because chemicals shape/influence our personalities and emotions. You'll see the same consistent behaviours in animals as you do in humans, when given the same chemicals.
Thats because chemicals shape/influence our personalities and emotions. You'll see the same consistent behaviours in animals as you do in humans, when given the same chemicals.
Dont know why this got down voted.
https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/RS01....
Your mobile phone can be used as a weapon against you, triggering calcium waves in the body which could be harmful to your liberty and your life.
I know because I've had it done to me!
https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/RS01....
Your mobile phone can be used as a weapon against you, triggering calcium waves in the body which could be harmful to your liberty and your life.
I know because I've had it done to me!
I often entertain the idea that all the patterns we observe are merely things that match our capability of understanding. This could explain the "unreasonable effectiveness of mathematics in the natural sciences".
I have a similar but somewhat different take on this. The universe comes first, it behaves - for what ever reasons - in the way it does. Than we humans show up and invent logic, a set of rules that is useful to reason about our universe. It either rains or it doesn't makes sense in our universe. But the universe could potentially have been different, for example something like the many-worlds interpretation but where the inhabitants of that universe can experience all the branches. Their logic might say it rains and it doesn't.
Other ideas like objects, properties, space, time, causation or countability might also be influenced by the way our universe works and how we perceive it and they might be far from universal or useful across different possible universes. On top of that we than construct mathematics and explore what all the things are that we can build from those ideas and our laws of logic. It should probably not be especially surprising that we can on the one hand construct things that are not realized in our universe and on the other hand find structures that are useful for describing and understanding our universe.
I have a similar but somewhat different take on this. The universe comes first, it behaves - for what ever reasons - in the way it does. Than we humans show up and invent logic, a set of rules that is useful to reason about our universe. It either rains or it doesn't makes sense in our universe. But the universe could potentially have been different, for example something like the many-worlds interpretation but where the inhabitants of that universe can experience all the branches. Their logic might say it rains and it doesn't.
Other ideas like objects, properties, space, time, causation or countability might also be influenced by the way our universe works and how we perceive it and they might be far from universal or useful across different possible universes. On top of that we than construct mathematics and explore what all the things are that we can build from those ideas and our laws of logic. It should probably not be especially surprising that we can on the one hand construct things that are not realized in our universe and on the other hand find structures that are useful for describing and understanding our universe.
I suspect something opposite, that math comes before the universe because the universe needs math but math doesn't need the universe. Math only calls for internal consistency which radically does not depend upon the universe. It describe perfectly self-consistent realities that do not exist. Math could model say for instance a world of continuous matter as opposed to our mostly empty matter for instance. We have calculated numbers larger than the universe itself could render after all.
I beg to differ. Math typically requires abstract thought, symbols, and humans to produce and enjoy it. Especially the latter is quite a dependency.
We could reduce the requirements down to an implementation of a Turing machine, or something similar. (For the argument I simply ignore whether the machine is conscious -- that seems irrelevant in this context.)
That still requires some kind of discrete switch, which may seem fairly minimal to a human observer, but in reality consists of tens of thousands of atoms to operate. Atoms used to be simple, but turn out to be quite complex as well.
Representing, say, a circle in a fairly minimal system such as this would probably require the cooperation of millions of atoms.
We could reduce the requirements down to an implementation of a Turing machine, or something similar. (For the argument I simply ignore whether the machine is conscious -- that seems irrelevant in this context.)
That still requires some kind of discrete switch, which may seem fairly minimal to a human observer, but in reality consists of tens of thousands of atoms to operate. Atoms used to be simple, but turn out to be quite complex as well.
Representing, say, a circle in a fairly minimal system such as this would probably require the cooperation of millions of atoms.
If universe was total chaos, then we would observe it, because absence of chaos isn't required by anthropic principle, but instead we observe universe to have at most pseudorandom processes, therefore universe isn't total chaos.
My point is that we as humans can observe only a (highly structured) part of the chaos. I am merely assuming a chaotic universe, because that seems more reasonable than an empty universe or an ordered universe.
(Yes I know that one cannot use "reasonable" as an argument for a context in which human arguments do not apply. The entire thing is a thought experiment, nothing more.)
(Yes I know that one cannot use "reasonable" as an argument for a context in which human arguments do not apply. The entire thing is a thought experiment, nothing more.)
Isn't this just the anthropic principle from a different angle?
I get the appeal, it's a minimalism thing. The thinking mind makes up patterns and checks them against each other, there is no ultimate reality, blah blah blah. Besides, a mathematical ground truth would be a kind of transcendence, in that it's prior to human thought, which makes it uncomfortably close to ideas of God.
Still, I've come around basically to mathematical platonism. The structure is out there, we just happen to be smart enough to tease some of it out.
I have two arguments for this. The first is the very existence of long-standing problems, and their eventual resolution either way. For centuries we were able to wonder whether Fermat's last "theorem" (which was really a conjecture at the time) was actually true, and eventually Wiles came around with an extremely complicated proof and it was settled. And we do believe that math/logic is consistent enough that someone else couldn't just have followed a different train of thought and come up with a proof of the opposite. How does a strict intuitionist account for this kind of situation?
The second, and possibly deeper argument, has to do with structural equivalences. I've been out of the field for decades, but I know that a standard trick in academic math is to develop structural equivalences between disparate fields. You want to prove something in an area of math, but it's hard, so you prove that the whole structure of that subfield has a one-to-one correspondence with the structure of another subfield, and then prove the corresponding theorem in the other subfield, which happens to be easier (see: analytic number theory). Again, this sounds like exploring an existing territory, not like arbitrarily building thought-bridges here and there. The bridges are where they are, and if you try to build one where reality didn't put it, your proof will get nowhere.
An even stronger form of this is that, in advanced mathematics, all kinds of notions of universality appear all the time. One of the most famous is probably computability theory. Just using a few basic symbols (say, integers, first order logic and some additive operations), you get theories of varying power. But as soon as you hit a certain level of richness, bang, all of a sudden, you've hit computability. Your theory is rich enough to embed a Turing machine, and therefore is exactly as rich and expressive as any other computable - even if one is based on numbers and multiplication, and the other on graphs or some such other weird thing.
Universality shows up in lots of places. I'm too far out of the field to remember them, but it starts with the very integer numbers - there are plenty of ways to formalize their initial construction, but the eventual result is exactly the same.
At this point my general thinking is that the bulk of the structure is pre-given. I have no special conjecture to make about how that comes to be - it's all a logical structure, prior to matter or thought, so unlike physics, it's not like there could be another universe out there with different basic mathematics.
Still, I've come around basically to mathematical platonism. The structure is out there, we just happen to be smart enough to tease some of it out.
I have two arguments for this. The first is the very existence of long-standing problems, and their eventual resolution either way. For centuries we were able to wonder whether Fermat's last "theorem" (which was really a conjecture at the time) was actually true, and eventually Wiles came around with an extremely complicated proof and it was settled. And we do believe that math/logic is consistent enough that someone else couldn't just have followed a different train of thought and come up with a proof of the opposite. How does a strict intuitionist account for this kind of situation?
The second, and possibly deeper argument, has to do with structural equivalences. I've been out of the field for decades, but I know that a standard trick in academic math is to develop structural equivalences between disparate fields. You want to prove something in an area of math, but it's hard, so you prove that the whole structure of that subfield has a one-to-one correspondence with the structure of another subfield, and then prove the corresponding theorem in the other subfield, which happens to be easier (see: analytic number theory). Again, this sounds like exploring an existing territory, not like arbitrarily building thought-bridges here and there. The bridges are where they are, and if you try to build one where reality didn't put it, your proof will get nowhere.
An even stronger form of this is that, in advanced mathematics, all kinds of notions of universality appear all the time. One of the most famous is probably computability theory. Just using a few basic symbols (say, integers, first order logic and some additive operations), you get theories of varying power. But as soon as you hit a certain level of richness, bang, all of a sudden, you've hit computability. Your theory is rich enough to embed a Turing machine, and therefore is exactly as rich and expressive as any other computable - even if one is based on numbers and multiplication, and the other on graphs or some such other weird thing.
Universality shows up in lots of places. I'm too far out of the field to remember them, but it starts with the very integer numbers - there are plenty of ways to formalize their initial construction, but the eventual result is exactly the same.
At this point my general thinking is that the bulk of the structure is pre-given. I have no special conjecture to make about how that comes to be - it's all a logical structure, prior to matter or thought, so unlike physics, it's not like there could be another universe out there with different basic mathematics.
The article seems poorly written, or at least self-contradictory in a way that makes me uncertain what it means. The introduction talks about the intuitionism framing mathematics as a human construction, in opposition to an objective reality. But the subsequent paragraph talks about the truth of proofs themselves as being subjective.
It seems to me that these are two different claims. I think mathematics is a human construction and doesn't have a real substance. Instead, mathematics is a system of assumptions and generative rules, and more generally a discipline around creating and operating such systems. But "truth" within a system of assumptions and generative rules is not subjective, it's mechanically provable.
A possible confusion remains in what "true" means. Can something relating to an imaginary system can be true, or is it false because truth can only apply to objective reality? I think it's trivially true. Gollum was a hobbit, within the system of Middle Earth. If it's not true, then we need a different word for true that does mean this, because this is what most people mean when they use the word about all sorts of imaginary constructs, from institutions to cultural symbols.
It seems to me that these are two different claims. I think mathematics is a human construction and doesn't have a real substance. Instead, mathematics is a system of assumptions and generative rules, and more generally a discipline around creating and operating such systems. But "truth" within a system of assumptions and generative rules is not subjective, it's mechanically provable.
A possible confusion remains in what "true" means. Can something relating to an imaginary system can be true, or is it false because truth can only apply to objective reality? I think it's trivially true. Gollum was a hobbit, within the system of Middle Earth. If it's not true, then we need a different word for true that does mean this, because this is what most people mean when they use the word about all sorts of imaginary constructs, from institutions to cultural symbols.
> I think mathematics is a human construction and doesn't have a real substance. Instead, mathematics is a system of assumptions and generative rules, and more generally a discipline around creating and operating such systems. But "truth" within a system of assumptions and generative rules is not subjective, it's mechanically provable.
Mathematics is a human construction, but it certainly has a real substance. What does "mechanically provable" even mean, if there is no absolute truth? Do you believe in the definition of a proof or not? Do you believe that whenever the assumptions of your theorem are true, and you have a proof of a conclusion, that then the conclusion is also true? If you do believe that, that's your absolute truth, then. If you don't believe that, a proof is meaningless, isn't it?
Mathematics is a human construction, but it certainly has a real substance. What does "mechanically provable" even mean, if there is no absolute truth? Do you believe in the definition of a proof or not? Do you believe that whenever the assumptions of your theorem are true, and you have a proof of a conclusion, that then the conclusion is also true? If you do believe that, that's your absolute truth, then. If you don't believe that, a proof is meaningless, isn't it?
I think it's a system of symbols and rules. By mechanically provable, I mean that given axioms (assumptions) and rules, you can devise a machine (i.e. something which follows rules, with no independent thinking or homunculus) which generates statements which follow from the axioms and rules, and this is what "true" means in the system.
So would you say that your system of symbols and rules is real?
Could it be that we both use the same system of symbols and rules, with the same assumptions, but derive different conclusions? If not, why not?
It has no substance other than its representations; there's nothing of it you can touch which is physical. At best, there is a correspondence between the physical and the system.
I agree with you here, at least in the sense that the system is definitely not physical. You didn't answer my question, though. Is it real?
All information that exists is physical, encoded in a physical substrate. Beads in an abacus, holes in a punched card, distributions of charge in a computer memory. Hypothetical information that is not encoded physically cannot be causal. A book or computer program that have not been written can have no effects. Only information that exists in a physical encoding can be causal, by virtue of it’s physicality.
Not sure where you are going with this. What is causality? Is there a non-mathematical way of making it precise? And if I have found some way to make it precise, does it matter if I write it down here in this HN comment, or on a piece of paper, or just think it? Is the mathematical content different depending on how it is expressed in physical reality?
if the system is sound, then (I think) by definition you cannot prove different (wrong) conclusions.
if you derive a different result, by soundness those would be equivalent ???
if you derive a different result, by soundness those would be equivalent ???
It does not really matter if the system is sound or not, right? Although of course a sound one is far more interesting. Anyway, any way of justifying this is mathematical (and so would be the definition of soundness, if it was relevant here). If math is not real, then there is no justification.
math describes (fragments) of reality;
therefore it is of no consequence if math as itself is "real" or not. it is intended to model whatever "real" even is.
somewhat similarly: in modern logical theories whatever "true" (and/including "false") even mean doesn't matter. is left out of the logical theory and it is effectively a mere parameter.
all the subject does is gurantee "truth in, truth out" (and complementarily "false in, false out")
the precise details of true "and/including" false, seems to me, are somewhere in the boundary between "classical" and "intuitionism" (or "constructivism")
the subtle distinction between intuitionism" and "constructivism" is above my pay grade (and seemingly above the paygrade of everybody I've had the chance of discussing this with)
therefore it is of no consequence if math as itself is "real" or not. it is intended to model whatever "real" even is.
somewhat similarly: in modern logical theories whatever "true" (and/including "false") even mean doesn't matter. is left out of the logical theory and it is effectively a mere parameter.
all the subject does is gurantee "truth in, truth out" (and complementarily "false in, false out")
the precise details of true "and/including" false, seems to me, are somewhere in the boundary between "classical" and "intuitionism" (or "constructivism")
the subtle distinction between intuitionism" and "constructivism" is above my pay grade (and seemingly above the paygrade of everybody I've had the chance of discussing this with)
> math describes (fragments) of reality;
This is only possible if math itself is real. Note that I am not saying that a particular axiom system like Euclidean geometry has some sort of "real physical manifestation". No, what I am saying is that logical reasoning itself is real. And our reasoning about logical reasoning is certainly real as well, even if logical reasoning itself happens in very abstract form. Math itself might be viewed by some as just a game of symbols. But that doesn't change the fact that the game itself is real. Would it be otherwise, then math would be about as important as chess.
This is only possible if math itself is real. Note that I am not saying that a particular axiom system like Euclidean geometry has some sort of "real physical manifestation". No, what I am saying is that logical reasoning itself is real. And our reasoning about logical reasoning is certainly real as well, even if logical reasoning itself happens in very abstract form. Math itself might be viewed by some as just a game of symbols. But that doesn't change the fact that the game itself is real. Would it be otherwise, then math would be about as important as chess.
I like to draw a distinction between real and ideal.
I insist that math is ideal. it models reality ideally.
this distinction is important because otherwise we mix together something, and the ideas and concepts (e.g. symbols and rules) we use to describe and model said something.
the game is not real. people playing the game are real, the game getting played is real. the game on its own as may be described in symbols is ideal.
i suppose what it all is all about is the intersection between this reality and this ideality.
I insist that math is ideal. it models reality ideally.
this distinction is important because otherwise we mix together something, and the ideas and concepts (e.g. symbols and rules) we use to describe and model said something.
the game is not real. people playing the game are real, the game getting played is real. the game on its own as may be described in symbols is ideal.
i suppose what it all is all about is the intersection between this reality and this ideality.
You can say that a certain axiom system models a certain part of reality in an ideal way. But whatever is ideal, is also real, because otherwise there is nothing that could model anything. So your intersection of reality and ideality is just ideality itself.
[deleted]
Wow, you excavated an ocean to cover a puddle. how this: "something which follows rules, with no independent thinking or homunculus" can be easier to prove and reason than the initial rules?
For example, what is simpler to reason out: does a chess move violate chess rules, or, there exist a method to construct an electomechanical device, that will Correctly determine whether this chess move is legal?
For example, what is simpler to reason out: does a chess move violate chess rules, or, there exist a method to construct an electomechanical device, that will Correctly determine whether this chess move is legal?
The reason I brought up a machine explain something as being mechanical is to clarify that it doesn't require intuition.
We can make machines that count. A trivial example: pebbles in a bucket. Neither the pebbles nor the bucket need intelligence to act as (have a correspondence with) a counter.
We can make machines that count. A trivial example: pebbles in a bucket. Neither the pebbles nor the bucket need intelligence to act as (have a correspondence with) a counter.
Completely agree about the machines. Many mathematical results become more rigorous as a result of "can be done on this kind of machine" type of proofs. However, if the goal is getting rid of intuition, machines don't help! Because no matter what the machine (bucket of pebbles or a Buchholz hydra), it takes a lot of intuition to prove, that particular machine correctly enforces intended rules. Usually more intuition, than the original problem.
Without having proof for the machine itself that machine is declared axiomatic. It is a valid way to go about things of course, but I would hesitate calling it "not requiring intuition".
Without having proof for the machine itself that machine is declared axiomatic. It is a valid way to go about things of course, but I would hesitate calling it "not requiring intuition".
>What does "mechanically provable" even mean, if there is no absolute truth? Do you believe in the definition of a proof or not?
Different Axioms lead to different provable statements.
Believing in standard mathematics basically means that you can not believe in absolute truth. Unless you also believe that some guys a hundred years ago figured the sole and completely perfect rules which totally correspond to reality.
Different Axioms lead to different provable statements.
Believing in standard mathematics basically means that you can not believe in absolute truth. Unless you also believe that some guys a hundred years ago figured the sole and completely perfect rules which totally correspond to reality.
> Believing in standard mathematics basically means that you can not believe in absolute truth.
I agree that if one follows an axiomatic approach strictly and consider "truth" to be a shorthand for "provable from in some logic from some set of non-logical axioms" [1] then one is rejecting any notion absolute truth, since everything is relative to some set of axioms, but I don't agree with the charactedisation of this as "standard"; it seems to me to be a very Formalist stance.
I'd argue that most mathenaticians consider themselves Platonists, and believe that the mathematical objects they are describing are real enough to form some kind of metamathematical "standard model", and "absolute truth" can be defined in the model-theoretic sense relative to this standard model, even if this is somewhat unavoidably handwavy.
[1] : Even if you do think this, "truth" is generally used by logicians in the model-theoretic sense of "truth in some specific model/structure compatible with the language".
I agree that if one follows an axiomatic approach strictly and consider "truth" to be a shorthand for "provable from in some logic from some set of non-logical axioms" [1] then one is rejecting any notion absolute truth, since everything is relative to some set of axioms, but I don't agree with the charactedisation of this as "standard"; it seems to me to be a very Formalist stance.
I'd argue that most mathenaticians consider themselves Platonists, and believe that the mathematical objects they are describing are real enough to form some kind of metamathematical "standard model", and "absolute truth" can be defined in the model-theoretic sense relative to this standard model, even if this is somewhat unavoidably handwavy.
[1] : Even if you do think this, "truth" is generally used by logicians in the model-theoretic sense of "truth in some specific model/structure compatible with the language".
>but I don't agree with the charactedisation of this as "standard"
I used it as an objective term, defining mathematical objects on terms of ZFC and truth being relative only to ZFC is the standard mathematical foundation. If you ask a random mathematician what he thinks the foundations of mathematics are it will most likely be ZFC, even if he disagrees with it on any level, it is still what he would set his rival theory against.
I used it as an objective term, defining mathematical objects on terms of ZFC and truth being relative only to ZFC is the standard mathematical foundation. If you ask a random mathematician what he thinks the foundations of mathematics are it will most likely be ZFC, even if he disagrees with it on any level, it is still what he would set his rival theory against.
Working, or at least claiming to work, in ZFC is fairly standard, but that doesn’t make it the definition of mathematical truth.
As a sibling comment mentioned, most mathematicians have a sense of truth that is not bound to any axiom system.
I don’t think it’s contradictory to work in ZFC whilst simultaneously having a non-axiomatic notion of mathematical truth.
I would hazard a guess that most (all?) working research mathematicians would accept the truth of the Gödel sentence for their preferred axiom system (and deductive calculus), be it ZF/ZFC or TG or something else entirely, so I cannot accept the claim that they see the “standard” notion of truth as being relative to all models of some axiom system.
You might think this is just nit-picking, but if it’s fine (in the sense that this is still “standard”) to add an arbitrarily large set of Pi_1 formulae to ZFC from repetitions of Gödel 1, then I don’t think we can say that ZFC is the standard basis of mathematical truth because this cannot be justified in ZFC; there must be some other (standard) notion of mathematical truth used to justify this.
As a sibling comment mentioned, most mathematicians have a sense of truth that is not bound to any axiom system.
I don’t think it’s contradictory to work in ZFC whilst simultaneously having a non-axiomatic notion of mathematical truth.
I would hazard a guess that most (all?) working research mathematicians would accept the truth of the Gödel sentence for their preferred axiom system (and deductive calculus), be it ZF/ZFC or TG or something else entirely, so I cannot accept the claim that they see the “standard” notion of truth as being relative to all models of some axiom system.
You might think this is just nit-picking, but if it’s fine (in the sense that this is still “standard”) to add an arbitrarily large set of Pi_1 formulae to ZFC from repetitions of Gödel 1, then I don’t think we can say that ZFC is the standard basis of mathematical truth because this cannot be justified in ZFC; there must be some other (standard) notion of mathematical truth used to justify this.
>Working, or at least claiming to work, in ZFC is fairly standard
Which is why I called it "standard".
> I don’t think we can say that ZFC is the standard basis of mathematical truth
You literally just said that working in ZFC is "standard". A standard is a social agreement, standards can be completely false and absurds, while being standards.
>As a sibling comment mentioned, most mathematicians have a sense of truth that is not bound to any axiom system.
Which I agree with.
Which is why I called it "standard".
> I don’t think we can say that ZFC is the standard basis of mathematical truth
You literally just said that working in ZFC is "standard". A standard is a social agreement, standards can be completely false and absurds, while being standards.
>As a sibling comment mentioned, most mathematicians have a sense of truth that is not bound to any axiom system.
Which I agree with.
You said:
> truth being relative only to ZFC is the standard mathematical foundation
I think this is at odds with:
> most mathematicians have a sense of truth that is not bound to any axiom system
Re.:
> You literally just said that working in ZFC is "standard"
Nowhere did I say that considering ZFC to be the arbiter of mathematical truth is standard, in fact I’m claiming the opposite.
Deciding to adopt a particular set of axioms as standard just means that I’ll accept a proof from those axioms without question; it doesn’t mean that I believe mathematical truth is precisely that which is a syntactic consequence of ZFC/TG/whatever.
Perhaps I’m misinterpreting your claim? At the moment I’m reading it as: “The belief of the majority of working research mathematicians is that mathematical truth is defined relative to ZFC.”
> truth being relative only to ZFC is the standard mathematical foundation
I think this is at odds with:
> most mathematicians have a sense of truth that is not bound to any axiom system
Re.:
> You literally just said that working in ZFC is "standard"
Nowhere did I say that considering ZFC to be the arbiter of mathematical truth is standard, in fact I’m claiming the opposite.
Deciding to adopt a particular set of axioms as standard just means that I’ll accept a proof from those axioms without question; it doesn’t mean that I believe mathematical truth is precisely that which is a syntactic consequence of ZFC/TG/whatever.
Perhaps I’m misinterpreting your claim? At the moment I’m reading it as: “The belief of the majority of working research mathematicians is that mathematical truth is defined relative to ZFC.”
>I think this is at odds with:
I don't think so. Mathenatical statements are almost always framed in the context of ZFC. This does not contradict that mathematicians think ZFC is not an absolute truth.
>Perhaps I’m misinterpreting your claim? At the moment I’m reading it as: “The belief of the majority of working research mathematicians is that mathematical truth is defined relative to ZFC.”
All I am saying is that mathematicians are framing their results in the context of ZFC and that this makes it the "standard" theory. I think that statement is absolutely not controversial, even alternative theories are framed in opposition to ZFC.
I absolutely do not think that mathematicians believe that ZFC is "true", as in it is the one and only perfect set of axioms.
My initial argument (and I am sorry if that was unclear) was that IF you believe that mathematical truth is about formal derivations from axioms (ZFC would be such a theory, same as ZF or any variation) then either you have to say that there is one perfect system and all truth is relative to it alone or that there are multiple equally true, but incompatible, theories.
The "IF" is of course important and I don't think many mathematicians actually agree with the IF clause. I actually completely agree with: "Deciding to adopt a particular set of axioms as standard just means that I’ll accept a proof from those axioms without question; it doesn’t mean that I believe mathematical truth is precisely that which is a syntactic consequence of ZFC/TG/whatever." and I am sorry if I wasn't clear. I actually do not think there is any disagreement here.
I don't think so. Mathenatical statements are almost always framed in the context of ZFC. This does not contradict that mathematicians think ZFC is not an absolute truth.
>Perhaps I’m misinterpreting your claim? At the moment I’m reading it as: “The belief of the majority of working research mathematicians is that mathematical truth is defined relative to ZFC.”
All I am saying is that mathematicians are framing their results in the context of ZFC and that this makes it the "standard" theory. I think that statement is absolutely not controversial, even alternative theories are framed in opposition to ZFC.
I absolutely do not think that mathematicians believe that ZFC is "true", as in it is the one and only perfect set of axioms.
My initial argument (and I am sorry if that was unclear) was that IF you believe that mathematical truth is about formal derivations from axioms (ZFC would be such a theory, same as ZF or any variation) then either you have to say that there is one perfect system and all truth is relative to it alone or that there are multiple equally true, but incompatible, theories.
The "IF" is of course important and I don't think many mathematicians actually agree with the IF clause. I actually completely agree with: "Deciding to adopt a particular set of axioms as standard just means that I’ll accept a proof from those axioms without question; it doesn’t mean that I believe mathematical truth is precisely that which is a syntactic consequence of ZFC/TG/whatever." and I am sorry if I wasn't clear. I actually do not think there is any disagreement here.
Yet still professional mathematicians have an underlying notion of truth outside of any axiom systems. I forgot who said it but if we were to find a contradiction using Peano’s axioms, we would say that the axioms were wrong, rather than arithmetic itself.
Even your comment references “perfect rules which totally correspond to reality” which seems to be another way to say “absolute truth”.
Even your comment references “perfect rules which totally correspond to reality” which seems to be another way to say “absolute truth”.
>Yet still professional mathematicians have an underlying notion of truth outside of any axiom systems.
I am certain about that, but it does not make my statement less true. I actually think that very few people believe in ZFC as either a formalist absolute or as an arbitrary set of rules. I think the most common view is that it enables other theories, that those mathematicians actually care about. The moment those theories rely directly on axioms things get difficult. I think the following quote describes quite well the state of ZFC: "The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"
I am certain about that, but it does not make my statement less true. I actually think that very few people believe in ZFC as either a formalist absolute or as an arbitrary set of rules. I think the most common view is that it enables other theories, that those mathematicians actually care about. The moment those theories rely directly on axioms things get difficult. I think the following quote describes quite well the state of ZFC: "The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"
I think the Stanford Encyclopedia of Philosophy article is better.
https://plato.stanford.edu/entries/intuitionism/
https://plato.stanford.edu/entries/intuitionism/
Yeah, that's much better.
But the subsequent paragraph talks about the truth of proofs themselves as being subjective.
This reflects the development of intuitionism. The project was first started by L. E. J. Brouwer, who rejected formalism. It was then developed by his student Arend Heyting, who formalized it with the Heyting algebra, a restricted variant of Boolean algebra that lacks the double negation elimination (~~p => p) and law of the excluded middle (p OR ~p).
Pretty much all work in intuitionistic mathematics continues the work of Heyting. Brouwer would have rejected the entire enterprise as subjective, so he is mainly of historical and philosophical interest.
This reflects the development of intuitionism. The project was first started by L. E. J. Brouwer, who rejected formalism. It was then developed by his student Arend Heyting, who formalized it with the Heyting algebra, a restricted variant of Boolean algebra that lacks the double negation elimination (~~p => p) and law of the excluded middle (p OR ~p).
Pretty much all work in intuitionistic mathematics continues the work of Heyting. Brouwer would have rejected the entire enterprise as subjective, so he is mainly of historical and philosophical interest.
>mathematics is a system of assumptions and generative rules, and more generally a discipline around creating and operating such systems.
This is essentially the standard mathematical approach developed during the early last century. From a few basic axioms (which are not really justifiable) new statements are proven and structures are built up. All notions of truth and provability are relative to that system. In standard ZFC (those are the standard axioms) mathematics "1+1=2" is, like all other statements, a statement about sets. The statement is true by the definitions of 1,2, "=" and "+". In an alternative system with different axioms or definitions the statement is false.
This is not the view of intuitionists though. For them the symbols 1,2 or + aren't formalized objects (e.g. sets in this case). They are just symbols, which transfer some (hopefully) shared meaning to another person, who then (potentially with some addition arguments) might also accept the truth of that statement.
In the former the question of "truth" is fully formal there can be no "interpretation" in any meaningful sense. Intuitionism places mathematics fully inside the mind of the mathematician and "truth" can only be found there.
>Can something relating to an imaginary system can be true, or is it false because truth can only apply to objective reality?
In formalized mathematics "truth" is fully formal. There can not be any "external" truth derived from it, as any such statement is non-sensical.
This is essentially the standard mathematical approach developed during the early last century. From a few basic axioms (which are not really justifiable) new statements are proven and structures are built up. All notions of truth and provability are relative to that system. In standard ZFC (those are the standard axioms) mathematics "1+1=2" is, like all other statements, a statement about sets. The statement is true by the definitions of 1,2, "=" and "+". In an alternative system with different axioms or definitions the statement is false.
This is not the view of intuitionists though. For them the symbols 1,2 or + aren't formalized objects (e.g. sets in this case). They are just symbols, which transfer some (hopefully) shared meaning to another person, who then (potentially with some addition arguments) might also accept the truth of that statement.
In the former the question of "truth" is fully formal there can be no "interpretation" in any meaningful sense. Intuitionism places mathematics fully inside the mind of the mathematician and "truth" can only be found there.
>Can something relating to an imaginary system can be true, or is it false because truth can only apply to objective reality?
In formalized mathematics "truth" is fully formal. There can not be any "external" truth derived from it, as any such statement is non-sensical.
I agree and this is actually a common problem in philosophical discussions. People struggle to differentiate between two kinds of objectivity: the "total" objectivity of knowledge that is completely context-free and unstructured, and the objective modulo a given set of assumptions (e.g., "the human brain", or at least, the human way of structuring and understanding reality). Mathematical truths are objective with respect to the latter kind of objectivity, but not the former.
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The older I get the more I feel like pursuing these trains of thought will lead to mental instability and sadness.
We model the world and the world follows the model and the exceptions add to our refinement of the model and that’s what makes it true.
Perception and reality are yin and yang and that is about as deep as my philosophy needs to go here
We model the world and the world follows the model and the exceptions add to our refinement of the model and that’s what makes it true.
Perception and reality are yin and yang and that is about as deep as my philosophy needs to go here
I understand where you're coming from, and pragmatic perspectives certainly have their place. However, philosophy, including intuitionism, can offer us new ways to comprehend the world. It's not about destabilizing our understanding, but rather enriching it.
Although it can be challenging, this exploration can also bring greater depth to our perception and reality, much like the yin and yang you mentioned. Even if we don't adopt a new philosophy wholeheartedly, engaging with it can open us up to valuable insights.
Although it can be challenging, this exploration can also bring greater depth to our perception and reality, much like the yin and yang you mentioned. Even if we don't adopt a new philosophy wholeheartedly, engaging with it can open us up to valuable insights.
Sounds like GPT wrote this comment
An illustration of AI disrupting society in indirect ways. If this comment were in fact written by human hands, your perception of it is lessened by the mere possibility it wasn't.
and AI thinks the Constitution was written by AI: https://arstechnica.com/information-technology/2023/07/why-a...
I started being much happier about philosophy in general once I realized a very important truth: "unlike hard sciences, philosophy only affects philosophers"
For example the toplevel article says things like "independent existence in an objective reality".. this sounds important, right? Perhaps depending on the truth of the intuitionism, we might discover some facts about real world?
Nope. The whole "is intuitionism true" question only affects the person thinking about it. The whole books are written that basically quibble about dictionary definitions of "fundamental principles", "objective reality", "mental activity" and other complex words.
(This is especially visible in discussion of "consciousness": there are tons of texts about it, and yet none of them matter in any practical way. The practical applications -- generative text models, NLP, neuroscience, etc.. -- just ignore all the philosophic cruft)
For example the toplevel article says things like "independent existence in an objective reality".. this sounds important, right? Perhaps depending on the truth of the intuitionism, we might discover some facts about real world?
Nope. The whole "is intuitionism true" question only affects the person thinking about it. The whole books are written that basically quibble about dictionary definitions of "fundamental principles", "objective reality", "mental activity" and other complex words.
(This is especially visible in discussion of "consciousness": there are tons of texts about it, and yet none of them matter in any practical way. The practical applications -- generative text models, NLP, neuroscience, etc.. -- just ignore all the philosophic cruft)
Intuitionism has had a great impact on computer science and the design of many programming languages.
It's now widely known that an intuitionistic proof has computational content (through the curry-howard correspondence)
Those things have a real world impact
It's now widely known that an intuitionistic proof has computational content (through the curry-howard correspondence)
Those things have a real world impact
I have no doubt it had an impact on Curry personally, but it does not it has direct impact to on the the computer science. The Curry–Howard correspondence is formulated entirely in mathematical terms, and no philosophy is required to understand and use it.
For example, Robert Goddard, who invented first liquid-fueled rocket, became interested in space when he read Wells' "The War of the Worlds". So did Wells' work had an impact on rocket science? Certainly. Does it mean you should read "The War of the Worlds" if you want to build rockets? Probably not.
For example, Robert Goddard, who invented first liquid-fueled rocket, became interested in space when he read Wells' "The War of the Worlds". So did Wells' work had an impact on rocket science? Certainly. Does it mean you should read "The War of the Worlds" if you want to build rockets? Probably not.
The philosophy has an impact in the mental model and the learnability of the mathematical theories. Technically you could learn and use the mathematics without knowing the philosophy behind it but in practice almost nobody does that.
It's like knowing design patterns in OO: it's ultimately all software right? But software exists first in the mind of the programmer, and what's written in the IDE is just a shadow of what the programmer imagined. This makes one appreciate the point that there are things in software that exists entirely inside our heads
It's like knowing design patterns in OO: it's ultimately all software right? But software exists first in the mind of the programmer, and what's written in the IDE is just a shadow of what the programmer imagined. This makes one appreciate the point that there are things in software that exists entirely inside our heads
> "unlike hard sciences, philosophy only affects philosophers"
Studying history will very quickly dissuade you from this notion, over and over again. For the perhaps most striking example, consider how different the 20th century would have looked like if a certain philosopher named Marx hadn't tried to marry the works of another philosopher, Hegel, with economic theory.
Studying history will very quickly dissuade you from this notion, over and over again. For the perhaps most striking example, consider how different the 20th century would have looked like if a certain philosopher named Marx hadn't tried to marry the works of another philosopher, Hegel, with economic theory.
And yet, it was not Marx or Hegel who took over the country and started campaign on Red Terror. And Lenin did not need to cite Das Kapital to start Russian Civil War. (I have no doubt he was inspired by it though)
(It's an interesting thought experiment what would have happened if there were no Karl Marx. February Revolution would have happened anyway.. but would bolsheviks still exist? Would Lenin still do all the things he did anyway, or would he become a civil servant instead?)
And for the most striking example, consider how different the 20th century would have looked if Academy of Fine Arts Vienna would have accepted a certain applicant from Linz. Does this mean fine arts educational institutions are very important in the political life?
(It's an interesting thought experiment what would have happened if there were no Karl Marx. February Revolution would have happened anyway.. but would bolsheviks still exist? Would Lenin still do all the things he did anyway, or would he become a civil servant instead?)
And for the most striking example, consider how different the 20th century would have looked if Academy of Fine Arts Vienna would have accepted a certain applicant from Linz. Does this mean fine arts educational institutions are very important in the political life?
Lenin wasn't only "inspired" by Marx, he wasn't even the first Russian socialist. Marxism was simply the framework that a lot of the socialist revolutionaries at that time were operating in (including ones that Lenin ended up opposing later).
Sure, given the economic and political realities of Russia at that time, something might have happened even if Marx hadn't existed, but it might have looked very different.
Philosophy matters to the world because other people read philosophy and are inspired by it to change something about the world. The French Revolution, US independence, Latin American independence and so on were all undertaken in no small part by people who had read a lot and were trying to make some of their ideals reality.
Sure, given the economic and political realities of Russia at that time, something might have happened even if Marx hadn't existed, but it might have looked very different.
Philosophy matters to the world because other people read philosophy and are inspired by it to change something about the world. The French Revolution, US independence, Latin American independence and so on were all undertaken in no small part by people who had read a lot and were trying to make some of their ideals reality.
There’s a whole lot more stuff in math than models of the world. Arguably, the vast majority of math has as much to do with the world as chess.
In the philosophy of mathematics, intuitionism
is an approach where mathematics is considered
to be purely the result of the constructive
mental activity of humans rather than the
discovery of fundamental principles claimed
to exist in an objective reality
When we say (or think) "Joe and Sue went to the grocery", it seems inherent to this statement that there are two distinct actors. Joe and Sue. But that is already math, isn't it? I have the feeling to avoid math, we would have to avoid "something" in the first place. As it already implies that "something" is in a category, consists of a collection of other things etc. So we could not talk about anything anymore.
I think both are descriptive. Mathematics is a descriptive language, and thoughts are descriptions of the world, ourselves, our intentions, etc or maybe the process of forming those descriptions.
>Wouldn't that mean that not only mathematics is pure mental activity, but every thought?
That statements seems very obviously true.
That statements seems very obviously true.
"pure mental activity" as opposed to "mental activity related to a reality outside of the mental activity".
I don't see how you could distinguish those. Even the purest form mathematics involves things like rules and symbols and those have to exist in reality (certainly they are at least as real as any emotion).
counterpoint:
remove all blackboards and chalk, and paper and pencils.
can you still do math?
remove all blackboards and chalk, and paper and pencils.
can you still do math?
Sure, it just takes a lot of remembering. But the point is that still that isn't "pure" mental activity. I am still imagining symbols and rules, those exist in reality at least as much as emotions do.
hmm, but imagining rules and symbols IS a pure mental activity, isn't it???
They are very similar to language, which cettainly is part of reality.
My first foray into math involved counting with my fingers.
This is post-modernism applied to math. It concludes in solipsism.
Since the mind and its mental constructs are a part of the objective reality, they will end up describing aspects of objective reality. If they don't, they break down, become chaotic and incomprehensible to those grounded in the objective reality.
Since the mind and its mental constructs are a part of the objective reality, they will end up describing aspects of objective reality. If they don't, they break down, become chaotic and incomprehensible to those grounded in the objective reality.
They don’t need to describe aspects of reality, they just need to describe something analogous that’s useful enough when applied.
I’m no physicist, but I understand that Newtonian physics aren’t strictly true as such, but they are a good enough analogy to put a person on a different planetary body.
So I think it’s fine to be agnostic and practical about the outcomes without needing say much about the metaphysics either way.
Anyway, both perspectives tickle my curiosity.
I’m no physicist, but I understand that Newtonian physics aren’t strictly true as such, but they are a good enough analogy to put a person on a different planetary body.
So I think it’s fine to be agnostic and practical about the outcomes without needing say much about the metaphysics either way.
Anyway, both perspectives tickle my curiosity.
For sure.
I'm of the "all models are wrong, some models are useful" school of thought. My best guess is that the platonic-ideals-are-real folks are mistaking something in their head for something outside it. That's not to deny that there is an objective reality out there, just that I have no particular reason to think that it's perfectly representable in 3 pounds of primate headmeat and expressible by squirting air through our meat-flaps. [1]
But ultimately, it doesn't matter too much to me, because the practical utility of both models is pretty high. It does make me wish to meet intelligent beings from different evolutionary backgrounds, though, as I think there would be a lot of "So you think what exactly?" that would be very revealing about which things are pan-human quirks and which are more universal.
[1] Credit goes to Terry Bisson here for the last bit: https://www.mit.edu/people/dpolicar/writing/prose/text/think...
I'm of the "all models are wrong, some models are useful" school of thought. My best guess is that the platonic-ideals-are-real folks are mistaking something in their head for something outside it. That's not to deny that there is an objective reality out there, just that I have no particular reason to think that it's perfectly representable in 3 pounds of primate headmeat and expressible by squirting air through our meat-flaps. [1]
But ultimately, it doesn't matter too much to me, because the practical utility of both models is pretty high. It does make me wish to meet intelligent beings from different evolutionary backgrounds, though, as I think there would be a lot of "So you think what exactly?" that would be very revealing about which things are pan-human quirks and which are more universal.
[1] Credit goes to Terry Bisson here for the last bit: https://www.mit.edu/people/dpolicar/writing/prose/text/think...
No, you're reading too much into it. The origin of the idea was a skepticism around seemingly paradoxical mathematical constructions, like uncountable infinities. Intuitionism eliminates some methods of proving that such things exist without needing to construct a proof of their existence.
And via Curry-Howard, any intuitionistic proof is also a computer program. Intuitionism thus unifies computation and mathematics in a very direct way, which has been extremely useful.
And via Curry-Howard, any intuitionistic proof is also a computer program. Intuitionism thus unifies computation and mathematics in a very direct way, which has been extremely useful.
Intuitionism is not at all chaotic, though. It's a fully cogent way of doing math, it's just not a perfect overlap with traditional mathematics: some things you can prove intuitionistically that you cannot prove otherwise, and vice versa.
I was initially surprised to read this because when I hear Intuitionism, I hear Intuitionist Logic. But IL doesn't have anything to do with denying objective reality; it can use facts on the way to proof. So I don't really know why Intuitionism is so much more adamant about denying constructive reality, or why it's thought to "give rise" to Intuitionist Logic, which at this point seems like a totally different thing. In other words, it's true that truth != proof, but that doesn't mean truth doesn't exist.
If there is an independent source of truth (external reality), then classical logic makes sense and intuitionistic logic doesn't. But intuitionists say mathematics, unlike the physical would, doesn't have such an independent reality. There is no platonic mathematical reality apart from explicit mathematical construction. Then classical logic is inappropriate and intuitionistic logic has to be used for mathematics.
So then what kind of logic is the following:
"Socrates is a male" has a value of True; "All men are mortal" has a value of True; and a conclusion of "Socrates is mortal" that is "True" because a True "and" a True yields a True.
And then an immortal man is discovered, making that first premise "False". And False "and" a True yields a "False".
This causes "Socrates is mortal" to have a truth value of "False", which doesn't make sense unless you consider it "proof" instead of "truth": it used to be proven (by the premises) that Socrates is mortal, but now it is "false that it is proven". It might still be true, but it's not proven.
This is closer to intuitionist logic than classical logic, but it still relies on facts like "Socrates is male".
"Socrates is a male" has a value of True; "All men are mortal" has a value of True; and a conclusion of "Socrates is mortal" that is "True" because a True "and" a True yields a True.
And then an immortal man is discovered, making that first premise "False". And False "and" a True yields a "False".
This causes "Socrates is mortal" to have a truth value of "False", which doesn't make sense unless you consider it "proof" instead of "truth": it used to be proven (by the premises) that Socrates is mortal, but now it is "false that it is proven". It might still be true, but it's not proven.
This is closer to intuitionist logic than classical logic, but it still relies on facts like "Socrates is male".
> "Socrates is a male" has a value of True; "All men are mortal" has a value of True; and a conclusion of "Socrates is mortal" that is "True" because a True "and" a True yields a True.
No, for a logical argument the conclusion must be true if the premises are true. A valid argument (proof) only shows: necessarily, if the premises are true, the conclusion is true.
> And then an immortal man is discovered, making that first premise "False". And False "and" a True yields a "False".
> This causes "Socrates is mortal" to have a truth value of "False"
No...
No, for a logical argument the conclusion must be true if the premises are true. A valid argument (proof) only shows: necessarily, if the premises are true, the conclusion is true.
> And then an immortal man is discovered, making that first premise "False". And False "and" a True yields a "False".
> This causes "Socrates is mortal" to have a truth value of "False"
No...
The offer of 'objective reality' as the antithesis to solipsistic mental constructs is exactly the naivety that give both of these impotent families of epistemology any continued sway. Both bad, both wrong, and the crowd swings from one to the other, and at each arrival, anew recognizes the flaws of the mode and turns back.
Thank god a philosopher has arrived to tell us we're all wrong. Being so wise, you must have the correct answer for us. What's it going to be today "you're not smart enough to understand my genius solution" or "enlightenment can't be taught, only achieved"?
Yeah. There are some approaches in philosophy of mathematics which try to avoid platonism (the view that mathematical objects have a mind independent existence) but while also retaining classical logic. That's not easily done though. (Currently popular is "structuralism", but this theory has its own problems.)
I do appreciate you providing some relevant information instead of just telling everyone else they're wrong. You still haven't put any of your own ideas on the line though. Is there an approach you believe is correct, and why?
Here's one of mine so I play by my own rules: philosophical questions don't have any testable hypothesis by nature, they'd be scientific questions if they did. The goal is to massage the question into an answerable one if you can. It's not really possible to have a "wrong" answer if the question is unknowable or malformed.
I could have made the original post with less melodrama but I like to snub philosophers when they swoop in like they have answers to these questions. Either it's a matter of science or you don't have answers, just perspectives.
Here's one of mine so I play by my own rules: philosophical questions don't have any testable hypothesis by nature, they'd be scientific questions if they did. The goal is to massage the question into an answerable one if you can. It's not really possible to have a "wrong" answer if the question is unknowable or malformed.
I could have made the original post with less melodrama but I like to snub philosophers when they swoop in like they have answers to these questions. Either it's a matter of science or you don't have answers, just perspectives.
> I do appreciate you providing some relevant information instead of just telling everyone else they're wrong. You still haven't put any of your own ideas on the line though. Is there an approach you believe is correct, and why?
I think a form of logicism (the view that mathematical statements are logical tautologies in disguise) is true at least for arithmetic. There are some ways to interpret them using logical truths from pure higher-order logic. I'm not sure about most other parts of mathematics.
> Here's one of mine so I play by my own rules: philosophical questions don't have any testable hypothesis by nature
I think they are testable by performing conceptual analysis, which consists in organizing data from semantic intuitions about the involved concepts.
I think a form of logicism (the view that mathematical statements are logical tautologies in disguise) is true at least for arithmetic. There are some ways to interpret them using logical truths from pure higher-order logic. I'm not sure about most other parts of mathematics.
> Here's one of mine so I play by my own rules: philosophical questions don't have any testable hypothesis by nature
I think they are testable by performing conceptual analysis, which consists in organizing data from semantic intuitions about the involved concepts.
What is testing a hypothesis?
If you'd like to ponder this question and more, please sign up for a course called "Philosophy of Science". It won't help answer it, but it's easy marks.
Sorry, I just figured from all your chest puffing that you had something to say, it appears you don't.
I made my point above. You're the one with nothing to say, you lead with a generic question.
Beautifully put.
I agree with this but I would also take it further. I think that all science is purely the mental activity of humans. What is really happening is that we are probing the nature of our own mind, even in fields like physics (and I say this as a physicist myself). My justification for this is Kantian. We can't get away from the fact that our brain structures reality so that we can understand it, for example by organising things via our notions of space and time. We can't access the "noumena", i.e. the things as they truly are, independent of our perception of them. Therefore the study of physics, or indeed literally any activity, can only ever be transcendental and reflexive, rather than actually reaching out into the "objective world".
Intuitionism is very interesting, but those of us who "grew up" on classical logic can easily accidentally depend on the law of excluded middle applying in all cases.
The Metamath system lets you specify the axioms you want to use, and then can verify that your proofs only use those axioms (directly or indirectly). There's a Metamath database specifically for intuitionistic logic:
https://us.metamath.org/ileuni/mmil.html
More things have been proven using intuitionistic logic over time.
The Metamath system lets you specify the axioms you want to use, and then can verify that your proofs only use those axioms (directly or indirectly). There's a Metamath database specifically for intuitionistic logic:
https://us.metamath.org/ileuni/mmil.html
More things have been proven using intuitionistic logic over time.
Because humans never experience anything in and of itself, but only the output of the interaction between sensory data and a brain, literally everything is purely the result of human mental activity.
I would agree that everything we experience is a model of the world that we construct from sense data, interpreted by our sensory systems and cognitive faculties. Donald Hoffman is good on this and worth looking up, although I disagree with some of his conclusions.
That doesn't mean the external physical world doesn't exist, the information we use to construct that model must come from somewhere, and we can deduce that the source is a persistent and consistent one.
The philosopher Husserl said: “The tree plain and simple, the thing of nature, is as different as it can be from this perceived tree as such, which as perceptual meaning belongs to the perception, and that inseparably.”
He came up with the idea of the noema which is our experience of something, and noesis which is our conscious act of perception. For me, that's our act of interpretation of our sensory perceptions. Sometimes this all goes wrong and we construct a flawed model that does not correspond perfectly to actual external reality, such as when we are deceived by optical illusions, stage magic or just hallucinate. Fortunately we can test and correct our perceptions through action in the physical world.
I'm an out-and-out physicalist but I think he is quite correct, we must distinguish between our internal perception of things and how things actually are. Fortunately science is extremely powerful in this regard. It has allowed us to decouple our model of the world from the limitations of our perceptual system, and come up with rigorous models of reality such as Relativity and Quantum Mechanics that are not tied to direct interpretation by our perceptive systems.
That doesn't mean the external physical world doesn't exist, the information we use to construct that model must come from somewhere, and we can deduce that the source is a persistent and consistent one.
The philosopher Husserl said: “The tree plain and simple, the thing of nature, is as different as it can be from this perceived tree as such, which as perceptual meaning belongs to the perception, and that inseparably.”
He came up with the idea of the noema which is our experience of something, and noesis which is our conscious act of perception. For me, that's our act of interpretation of our sensory perceptions. Sometimes this all goes wrong and we construct a flawed model that does not correspond perfectly to actual external reality, such as when we are deceived by optical illusions, stage magic or just hallucinate. Fortunately we can test and correct our perceptions through action in the physical world.
I'm an out-and-out physicalist but I think he is quite correct, we must distinguish between our internal perception of things and how things actually are. Fortunately science is extremely powerful in this regard. It has allowed us to decouple our model of the world from the limitations of our perceptual system, and come up with rigorous models of reality such as Relativity and Quantum Mechanics that are not tied to direct interpretation by our perceptive systems.
I think there is a hard limit to what we know and what we can assume to know based of this point and in logic by the Münchhausen trilemma. It's interesting to think of the source of sense data as persistent or consistent when it could just be that our sense organs reduce varied data into persistent experience.
When we look at a tree, it could very well be that the source of the tree is very much like the tree we experience, but it could also be wildly different. When we see a tree in a video game, we know there is no real source tree just like it, just ones and zeroes. I disagree that science fixes this problem. Tools are still just measuring the physical world. For example, if you used a tool to measure some aspect of the tree, you are still measuring the representation of the tree in this world. If I use the video game analogy again, my point is that you wouldn't be able to see true underlying 'source code' of the game tree by looking at it in the game.
When we look at a tree, it could very well be that the source of the tree is very much like the tree we experience, but it could also be wildly different. When we see a tree in a video game, we know there is no real source tree just like it, just ones and zeroes. I disagree that science fixes this problem. Tools are still just measuring the physical world. For example, if you used a tool to measure some aspect of the tree, you are still measuring the representation of the tree in this world. If I use the video game analogy again, my point is that you wouldn't be able to see true underlying 'source code' of the game tree by looking at it in the game.
I agree certain knowledge may be unattainable, but I don’t care. Useful effective knowledge that helps me achieve my goals in life will do just fine. As long as my mental model of the tree is accurate and useful enough for me to chop it down and make a table out of it, I’m good.
I don’t expect any description to accord perfectly with the reality of the object it refers to. It’s just a description, which may be more or less accurate or useful. Science, and investigation in general, is a way to test and improve such descriptions.
I don’t expect any description to accord perfectly with the reality of the object it refers to. It’s just a description, which may be more or less accurate or useful. Science, and investigation in general, is a way to test and improve such descriptions.
This is the transcendental idealism of Kant. He makes a distinction between the noumena (things in themselves beyond human cognition) and phenomena (things as they appear to us through our senses). Our mind constructs "transcendental objects" which are merely abstract ideas based on the appearances.
I do not agree, humans experience some interactions with Turing machines "in and of itself", because sensory data becomes irrelevant. A bit is always a bit.
This gives an argument about intuitionism: If you say that math is a byproduct of our wetware and nothing else, how come we can successfully teach it to turing machines, and have that process fill us on some holes we had in our understanding of maths, but not terribly large holes?
This gives an argument about intuitionism: If you say that math is a byproduct of our wetware and nothing else, how come we can successfully teach it to turing machines, and have that process fill us on some holes we had in our understanding of maths, but not terribly large holes?
I've been thinking about this recently, and realised that your framing here casts humans as separate from the rest of reality. Your sense organs and your brain are part of things-in-and-and-of-themselves.
I don't think it does. Humans are agents within reality and have perceptions of reality. Your brain having a representation in this reality that might be different from 'true reality' doesn't change the argument at all.
I don't see how your perceptions can be anything other than a direct experience of reality interacting with itself unless you imagine that your mind is separated from reality somehow
We do have a direct experience with reality, but we are only capable of processing an approximate & infinitely simplified model of it.
When you take a picture of an apple, you have a picture of an apple, not the apple itself. Both are real and related, but not the same thing.
When you take a picture of an apple, you have a picture of an apple, not the apple itself. Both are real and related, but not the same thing.
Yeah, this was the big step from Kant to Hegel, the realisation that the object is actually totally inside the subject and vice versa. Unfortunately, when the subject and object get totally mixed up in that way, the philosophy seems to become much more difficult and complicated. Kant's Transcendental Idealism is really useful and easy to understand, but if you want to go a step further into what you describe then it's like moving from Newtonian gravity to General Relativity. Literally everything becomes way more difficult.
Presumably the mental activity is itself explained by independent external reality. Intuitionists say there is no such independent reality for mathematics.
That doesn't really hold when you start measuring, unless you believe that your eyes can change the size of a measure tape depending on the subject.
> unless you believe that your eyes can change the size of a measure tape depending on the subject
Well, of course, they kind of can. There are drugs that make the world look like it's squashed, such as ketamine. The way that we perceive reality really is totally dependent on the properties of the observer. Of course we all, with our sober minds, assert that we are perceiving the ruler the "right way", but all this means it that we perceive the ruler in a way that most humans agree with. Jumping from that to "this is the way that the ruler looks for all possible subjects" is a leap of faith.
Well, of course, they kind of can. There are drugs that make the world look like it's squashed, such as ketamine. The way that we perceive reality really is totally dependent on the properties of the observer. Of course we all, with our sober minds, assert that we are perceiving the ruler the "right way", but all this means it that we perceive the ruler in a way that most humans agree with. Jumping from that to "this is the way that the ruler looks for all possible subjects" is a leap of faith.
The law of excluded middle always seemed like BS to me from the moment it was taught. It's wonderful that by removing it, proofs become "harder" to make, but consequently constructive and thus more rigoris.
Too many people believe in the law of excluded middle, especially in their own lives, much to the folly of civilization itself.
Too many people believe in the law of excluded middle, especially in their own lives, much to the folly of civilization itself.
Even if you "believe" in LEM, I think it could be helpful to conceptualize proofs that use it as instead being proofs in the "Reader monad" over LEM. So instead of proving `A => B`, you prove `A => Reader[LEM, B] = A => (LEM => B)`. If you really examine the proof you're doing, probably you don't need LEM in its full power, but actually some specific `Not[Not[X]] => X` (or a handful of specific instances like that).
The neat thing about this is that in principle it could be tracked in a proof assistant with type inference. The ZIO framework for Scala has a super slick system for dependencies where e.g. an `RIO[Foo,A]` (an IO that requires a Foo and returns an A) and `A => RIO[Bar,B]` can compose to form an RIO[Foo&Bar, B] with the types inferred. So you could in principle have a proof system that lets you infer types like `A => Reader[LEM[Foo] & LEM[Bar], B]` (where LEM[T] = Not[Not[T]] => T), i.e. you get explicit types that show all of the instances of an LEM function you need to make your proof constructive, and since you're thinking of things as living in the Reader monad, you have the ergonomics of a proof that just assumes LEM.
Same idea could be used with AC or any other "controversial" axiom. So you don't need to "believe" them to use them, and if you do "believe" them, you can benefit from pretending you don't.
The neat thing about this is that in principle it could be tracked in a proof assistant with type inference. The ZIO framework for Scala has a super slick system for dependencies where e.g. an `RIO[Foo,A]` (an IO that requires a Foo and returns an A) and `A => RIO[Bar,B]` can compose to form an RIO[Foo&Bar, B] with the types inferred. So you could in principle have a proof system that lets you infer types like `A => Reader[LEM[Foo] & LEM[Bar], B]` (where LEM[T] = Not[Not[T]] => T), i.e. you get explicit types that show all of the instances of an LEM function you need to make your proof constructive, and since you're thinking of things as living in the Reader monad, you have the ergonomics of a proof that just assumes LEM.
Same idea could be used with AC or any other "controversial" axiom. So you don't need to "believe" them to use them, and if you do "believe" them, you can benefit from pretending you don't.
It directly leads to the continuum being inseperable.
If you believe that "now" is real, you believe in LEM.
If you believe that "now" is real, you believe in LEM.
Well, I'm an eternalist who remembers the basics of the Relativity of simultaneity, so I reject that "now is real" as such.
Do you believe that any thing can be seperated into two part, which, if put together exactly the same as before become the initial part?
Paraphrasing a math professor I had [mumble] decades ago:
"In theory, mathematics is all pure & abstract logic, with no connection whatsoever to the real world.
"In practice, if you want funding for your mathematical research, or for more than a puny handful of people to ever look at what you did, then you had best pay plenty of attention to the real-world usefulness of it."
"In theory, mathematics is all pure & abstract logic, with no connection whatsoever to the real world.
"In practice, if you want funding for your mathematical research, or for more than a puny handful of people to ever look at what you did, then you had best pay plenty of attention to the real-world usefulness of it."
> In theory, mathematics is all pure & abstract logic, with no connection whatsoever to the real world.
I understand what your math professor was trying to communicate, but at the same time, I think framing things this way begs the question that mathematical entities are not "real". It's more accurate (or at least less question begging) to say something like "mathematics, or mathematical objects, doesn't seem to have any obvious or direct connection to the material or physical world". Putting things this way doesn't reify mathematical entities, but it also doesn't presume that they don't exist.
I understand what your math professor was trying to communicate, but at the same time, I think framing things this way begs the question that mathematical entities are not "real". It's more accurate (or at least less question begging) to say something like "mathematics, or mathematical objects, doesn't seem to have any obvious or direct connection to the material or physical world". Putting things this way doesn't reify mathematical entities, but it also doesn't presume that they don't exist.
I imagine that my old prof. would say something like: "Mathematicians agreed long ago on a very short and exacting definition for 'real' numbers, and got on with doing useful work. Philosophers never agree on short nor exacting definitions for anything, and certainly don't want to do anything useful."
>mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality
In the context of the Godel Incompleteness Theorem I have always found it difficult to reconcile mathematics w/ an objective reality. A mathematical system would simply be incapable of completely encompassing objective reality.
In the context of the Godel Incompleteness Theorem I have always found it difficult to reconcile mathematics w/ an objective reality. A mathematical system would simply be incapable of completely encompassing objective reality.
Gödel was a platonist btw.
What is the opposite of an objective reality?
I don’t know, but an objective reality could exist without it being fully accessible to us or directly corresponding to our mathematical systems.
I guess the opposite would be something like solipsism though.
I guess the opposite would be something like solipsism though.
Solipsism assumes the objective reality of the self.
That would fit though, wouldn’t it? Objective reality == exists independently of the self. Solipsism == only the self is a given. At the very least it’s opposite-ish. And if not, then I simply don’t have a better answer, but it wasn’t my question to begin with.
It seems that it builds towards your questions though, and we would be better off if we could come to some agreement on objective reality.
Although I understand it’s unlikely philosophers will ever agree on what reality is.
Although I understand it’s unlikely philosophers will ever agree on what reality is.
There's something truly evil about psychologizing math.
If anyone feels compelled by this you should probably start by learning about the many rock-solid antipsychologistic arguments out there that are at least 130 years old:
https://plato.stanford.edu/entries/psychologism/#FreAntArg
If anyone feels compelled by this you should probably start by learning about the many rock-solid antipsychologistic arguments out there that are at least 130 years old:
https://plato.stanford.edu/entries/psychologism/#FreAntArg
Not sure why you are getting downvoted, as I believe there is a kernel of truth to what you are saying. Although I am sympathetic to the idealist position, based on what we know today about computers, I have come to believe that the thing some would call mathematics today is not grounded in reality. Whenever mathematics departs from reality, it leads those who practice it down a path that ends with insanity.
In the same way that psychology can be harmful to the pursuit of truth, there is an evil or deformity in what pure mathematics represents (or became), as Cantor and Gödel both discovered trying to eliminate its paradoxes. Mathematics as practiced by, e.g., Leibniz and Euler, was much more interested in calculation, i.e., mechanization. It is this form of "mathematics", as the thing has come to be called, that should have been developed, instead of trying to axiomatize infinite sets, explain the continuum, or fix impredicativity.
But despite the many paradoxes of mathematics (e.g., AoC, LEM) and machines (i.e., bugs), computer science has happily chugged along and made many practical contributions which explain the universe and shed light into people's minds, fulfilling the role that pure mathematics once served. Today, we are very close to mechanizing both minds and mathematics, and can replicate many aspects of reality (i.e., mathematical or otherwise) inside computers. It is not unimaginable that one day, computer science as it is practiced today will be seen as a kind of telescope for revealing the nature of mathematics and/or reality, which the mind alone cannot fathom.
In the same way that psychology can be harmful to the pursuit of truth, there is an evil or deformity in what pure mathematics represents (or became), as Cantor and Gödel both discovered trying to eliminate its paradoxes. Mathematics as practiced by, e.g., Leibniz and Euler, was much more interested in calculation, i.e., mechanization. It is this form of "mathematics", as the thing has come to be called, that should have been developed, instead of trying to axiomatize infinite sets, explain the continuum, or fix impredicativity.
But despite the many paradoxes of mathematics (e.g., AoC, LEM) and machines (i.e., bugs), computer science has happily chugged along and made many practical contributions which explain the universe and shed light into people's minds, fulfilling the role that pure mathematics once served. Today, we are very close to mechanizing both minds and mathematics, and can replicate many aspects of reality (i.e., mathematical or otherwise) inside computers. It is not unimaginable that one day, computer science as it is practiced today will be seen as a kind of telescope for revealing the nature of mathematics and/or reality, which the mind alone cannot fathom.
Intuitionism has nothing to do with psychologizing math. The intuitionist is perfectly fine accepting that "the mind of the mathematician" is merely a metaphor for some system capable of mathematical constructions, and that the "mathematician" in this argument can be replaced by a computer performing the same steps. In fact, a lot of computer science, particularly programming language theory, depends on this very notion.
I think most here would know that math is not complete, consistent or decidable. (https://www.youtube.com/watch?v=HeQX2HjkcNo) But I'm going to leave that aside as it's pretty high level math for me and I never run into those problems in my life.
My personal problem with math that prevents me from seeing it as "discovery of fundamental principles claimed to exist in an objective reality" is natural numbers. It's impossible for me to clearly find the number 1 (for example) anywhere. The boundaries between one unit and 0.X or 1.Y seem arbitrary and chosen to help us create models. Religion and philosophy deal with this, but it's also apparent in digital signal processing. Or medicine: are the numerous bacteria in our bodies us, or not? Is the heat radiation from my body me? What about the fact that 1 + 1 rarely if ever equals 2? Meaning that two things together physically interact to create more than the sum of both parts. From celestial bodies to a replicated database the complexity goes through the roof when you have more than one thing.
My personal problem with math that prevents me from seeing it as "discovery of fundamental principles claimed to exist in an objective reality" is natural numbers. It's impossible for me to clearly find the number 1 (for example) anywhere. The boundaries between one unit and 0.X or 1.Y seem arbitrary and chosen to help us create models. Religion and philosophy deal with this, but it's also apparent in digital signal processing. Or medicine: are the numerous bacteria in our bodies us, or not? Is the heat radiation from my body me? What about the fact that 1 + 1 rarely if ever equals 2? Meaning that two things together physically interact to create more than the sum of both parts. From celestial bodies to a replicated database the complexity goes through the roof when you have more than one thing.
That's not quite correct. Systems of mathematics cannot be both complete and consistent, but incomplete systems of mathematics can be consistent. For example Presburger arithmetic is provably consistent. There are limits to consistency for sure, but that doesn't mean there's no such thing as mathematical consistency.
I'm not sure what you mean. Presburger arithmetic is famously complete. What a system can't be is consistent, complete, and strong enough to perform a Godel encoding (which requires something multiplication-like). Drop any of the three requirements and it's possible.
Inconsistent: trivial, from falsehood follows anything.
Incomplete: Peano.
Weak: Presburger.
Inconsistent: trivial, from falsehood follows anything.
Incomplete: Peano.
Weak: Presburger.
The comment could be interpreted as meaning that such systems cannot be complete or consistent, I'm just pointing out they can be one or the other. As I understand it, it is possible to consistently prove and decide things in mathematics, just not everything. Godel proved limits to mathematics, not that mathematics doesn't work. That's all.
> Godel proved limits to mathematics, not that mathematics doesn't work
Nobody claimed it doesn't work. It clearly does. The question is if it's a fundamental property of the universe or just an useful but flowed human mental model.
Nobody claimed it doesn't work. It clearly does. The question is if it's a fundamental property of the universe or just an useful but flowed human mental model.
Sure, personally I see it as a language for expressing relationships and processes. I expounded on this in detail in another comment.
> Inconsistent: trivial, from falsehood follows anything.
Only trivial if you accept the principle of explosion (ex falso quodlibet or ex contradictione quodlibet). If you reject it, you end up with paraconsistent logic, from which one can develop nontrivial inconsistent mathematics see https://plato.stanford.edu/entries/mathematics-inconsistent/ and https://ir.canterbury.ac.nz/bitstream/handle/10092/5626/1263...
Only trivial if you accept the principle of explosion (ex falso quodlibet or ex contradictione quodlibet). If you reject it, you end up with paraconsistent logic, from which one can develop nontrivial inconsistent mathematics see https://plato.stanford.edu/entries/mathematics-inconsistent/ and https://ir.canterbury.ac.nz/bitstream/handle/10092/5626/1263...
> Systems of mathematics cannot be both complete and consistent
No. They can't be at the same times complete, consistent, decidable and powerful enough to express arithmetic. You can do complete, consistent and decidable though.
No. They can't be at the same times complete, consistent, decidable and powerful enough to express arithmetic. You can do complete, consistent and decidable though.
>I think most here would know that math is not complete, consistent or decidable.
There is zero evidence ZFC is inconsistent.
Even if "1" does not exist in reality mathematics still describes fundamental universal principles. As long as you believe that these fundamental principles exist at all they exist as mathematical ones.
Not even hardcore Platonists would claim that the number 1 exists in physical reality. But that does not mean it doesn't exist in some abtract sense. You can construct Models of reality using the natural numbers and these models about real objects are just imperfect descriptions of reality.
There is zero evidence ZFC is inconsistent.
Even if "1" does not exist in reality mathematics still describes fundamental universal principles. As long as you believe that these fundamental principles exist at all they exist as mathematical ones.
Not even hardcore Platonists would claim that the number 1 exists in physical reality. But that does not mean it doesn't exist in some abtract sense. You can construct Models of reality using the natural numbers and these models about real objects are just imperfect descriptions of reality.
There's also zero conclusive evidence that ZFC is consistent. And even worse: if you found a proof (within ZFC or a weaker system) that ZFC was consistent, you would immediately know (by Gödel's second theorem) that it is actually inconsistent. The most we could hope for is that we couls prove its consistency in another system (one that hopefully convinces us more of its evident truth?).
ZFC is weird (especially choice). It's not implausible, but there's little a priori reason to assume that it describes some phyiscal reality. It just happens to give a foundation to a lot of really useful mathematics.
You could take a theory such as Peano Arithmetic and argue that that one is self-evident. But unfortunately, again by the second theorem, you can't use PA to prove ZFC consistent. That's, roughly, what Hilbert wanted to do in order to convince his critics, and he failed.
ZFC is weird (especially choice). It's not implausible, but there's little a priori reason to assume that it describes some phyiscal reality. It just happens to give a foundation to a lot of really useful mathematics.
You could take a theory such as Peano Arithmetic and argue that that one is self-evident. But unfortunately, again by the second theorem, you can't use PA to prove ZFC consistent. That's, roughly, what Hilbert wanted to do in order to convince his critics, and he failed.
Human thought is paraconsistent - relevance/relevant logic best models how implication works in natural language, and that is paraconsistent; I think the intelligibility of inconsistent fiction such as Graham Priest’s Sylvan’s Box [0] is also evidence of that. If one believes mathematics is ultimately grounded in human thought, and if human thought is ultimately paraconsistent, that suggests paraconsistent logic may be a better foundation for mathematics than classical logic. It also suggests that maybe we should seriously consider taking the inconsistency horn of Godel’s trilemma (incomplete or inconsistent or weak), given the paraconsistent rejection of the principle of explosion means that doing so is non-trivial. Inconsistent theories can be strong, complete and non-trivial.
[0] https://projecteuclid.org/journals/notre-dame-journal-of-for...
[0] https://projecteuclid.org/journals/notre-dame-journal-of-for...
I would bet that, even if ZFC is not consistent, there's another set of axioms which is, and in which all the stuff we've proven in ZFC still holds. That is, ZFC just happens to be a useful framework for the mathematics we're interested in. Even if ZFC collapses it seems very unlikely that all the stuff we've proven within it will; instead, we'll fix ZFC, like ZFC "fixed" naive set theory.
"1" works just fine for things that are discrete. Take eggs, for example. I have one egg. If it's smaller than other eggs, I don't have 0.9 eggs; no, I have exactly one egg or, if you prefer, I have exactly one small egg, but still exactly one.
I have exactly one wife. If she gained weight, I would not then have 1.01 wives.
I have exactly one cat. If she had N kittens, I would then have exactly N+1 cats, not (1 * N/10) cats.
And so on.
I have exactly one wife. If she gained weight, I would not then have 1.01 wives.
I have exactly one cat. If she had N kittens, I would then have exactly N+1 cats, not (1 * N/10) cats.
And so on.
My looking at natural numbers gave me the exact opposite conclusion.
Once the grey blob of infinite nothingness becomes distinct enough that there is a difference between one moment/point and another moment/point, you have enough to start counting different states (or do binary arithmetic). Its high school math to find out some numbers are different then others and eventually you'll find the primes.
I'll bet there is no god, power, or alternative rules in any possible universe, fictional or real, that could not find the primes.
Once the grey blob of infinite nothingness becomes distinct enough that there is a difference between one moment/point and another moment/point, you have enough to start counting different states (or do binary arithmetic). Its high school math to find out some numbers are different then others and eventually you'll find the primes.
I'll bet there is no god, power, or alternative rules in any possible universe, fictional or real, that could not find the primes.
> It's impossible for me to clearly find the number 1 (for example) anywhere.
what about that there is one of you?
what about that there is one of you?
> To an intuitionist, the claim that an object with certain properties exists is a claim that an object with those properties can be constructed
But, like, just in someone's mind ... to keep the truth of that existence purely subjective.
But, like, just in someone's mind ... to keep the truth of that existence purely subjective.
"My intuition has 10 significant digits of accuracy when it comes to calculating physical constants with quantumchromodynamics"
There's room for contemplation on the subjective nature of our perception and its influence on the understanding of reality
[deleted]
So why every human would agree that there are no squared circles?
The reality is that we can only have a sensible conversation about, or write sensibly about, concepts and facts within the context of our epistemic limit as a species.
You may have noticed that a lot of the cliché philosophical questions haven't really shifted for as long as people have been asking them. It isn't because there is no answer, but rather the question itself is null and void. It's answer is beyond our ability to find or compute because you would have remove yourself from the human frame of reference to find it.
The meaning of life? Doesn't make sense outside of the context of human life. In that context, it's whatever you want it to be or decide it is to you. Outside, well, you'd have to die to see if there's anything beyond, and as Spock points out in ST4, it would be impossible to discuss without a common frame of reference with someone who hasn't died.
~~~
To claim that math is a human invention or a noumenal language of nature is futile, the answer is not within our ability to determine. It is, for all intents and purposes, undefined. Unknowable.
That's not to say that the conversation is meaningless, as the logical positivists would claim, as this is also a leap too far.
Early Wittgenstein, big influence on the logical positivists, didn't try to claim that philosophy was meaningless, just that if you try to talk beyond the facts, you won't get anywhere.
Bradley aways summed it up nicely for me: "anyone ready to dispense with metaphysics is a brother metaphysician with a rival theory of his own"
In other words, to rule out metaphysics (defined as concepts and ideas that lack empirical data or basis in fact) is to make a metaphysical proposition, since by definition, there is no data to disprove the metaphysical propositions, just as much as there is no data to validate them.
Most metaphysical philosophies caught on because they can be presented rationally as a chain of propositions and conclusions, but they only add up within their own framework that is upheld by an unprovable assumption or set of assumptions.
Example, cogito ergo sum makes perfect sense, provided that you accept that the speaker is I, that the speaker thinks rather than simply utters, and that to be means anything at all. In the end, cogito ergo sum can be accepted intuitively within the framework of "let's not be sceptical of every thing" but what can you conclude from it?
"I think therefore I am" well, I is a thing that thinks
-> "I thinks, therefore it exists" I think presupposes I's existence
-> "I thinks"
That is a definition of I
I = thinks === I is a consciousness
which is a tautology. And that's why is makes sense, because it doesn't go anywhere than where it started, it's algebra.
It starts off as a = b where b = a and can be reduced to simply a
Math is privileged in that its algebraic statements can be used to model things in the world because it's numbers and functions of numbers, but ultimately the usefulness emerges from proving that a equals a very complicated statement that isn't obviously tautologically equal, or not equal to, a.
Math is a tool. It's a way of reasoning that comes bundled with reasonably standardised notation that enables boosted productivity compared to reasoning about the same problems in regular languages.
~~~
It's the same reason why I find "simulated reality" questions rather dull. If the world is a perfect simulation, we have no way to distinguish it from the real thing, and so to our frame of reference, there is no difference about which we can have a discussion that makes any sense.
If there are cracks in the simulation, sure, then you have a fact to talk about. But then the discussion isn't philosophical, it's practical: We've been brain-hacked, what do we do? and the conversation ends rather abruptly.
You may have noticed that a lot of the cliché philosophical questions haven't really shifted for as long as people have been asking them. It isn't because there is no answer, but rather the question itself is null and void. It's answer is beyond our ability to find or compute because you would have remove yourself from the human frame of reference to find it.
The meaning of life? Doesn't make sense outside of the context of human life. In that context, it's whatever you want it to be or decide it is to you. Outside, well, you'd have to die to see if there's anything beyond, and as Spock points out in ST4, it would be impossible to discuss without a common frame of reference with someone who hasn't died.
~~~
To claim that math is a human invention or a noumenal language of nature is futile, the answer is not within our ability to determine. It is, for all intents and purposes, undefined. Unknowable.
That's not to say that the conversation is meaningless, as the logical positivists would claim, as this is also a leap too far.
Early Wittgenstein, big influence on the logical positivists, didn't try to claim that philosophy was meaningless, just that if you try to talk beyond the facts, you won't get anywhere.
Bradley aways summed it up nicely for me: "anyone ready to dispense with metaphysics is a brother metaphysician with a rival theory of his own"
In other words, to rule out metaphysics (defined as concepts and ideas that lack empirical data or basis in fact) is to make a metaphysical proposition, since by definition, there is no data to disprove the metaphysical propositions, just as much as there is no data to validate them.
Most metaphysical philosophies caught on because they can be presented rationally as a chain of propositions and conclusions, but they only add up within their own framework that is upheld by an unprovable assumption or set of assumptions.
Example, cogito ergo sum makes perfect sense, provided that you accept that the speaker is I, that the speaker thinks rather than simply utters, and that to be means anything at all. In the end, cogito ergo sum can be accepted intuitively within the framework of "let's not be sceptical of every thing" but what can you conclude from it?
"I think therefore I am" well, I is a thing that thinks
-> "I thinks, therefore it exists" I think presupposes I's existence
-> "I thinks"
That is a definition of I
I = thinks === I is a consciousness
which is a tautology. And that's why is makes sense, because it doesn't go anywhere than where it started, it's algebra.
It starts off as a = b where b = a and can be reduced to simply a
Math is privileged in that its algebraic statements can be used to model things in the world because it's numbers and functions of numbers, but ultimately the usefulness emerges from proving that a equals a very complicated statement that isn't obviously tautologically equal, or not equal to, a.
Math is a tool. It's a way of reasoning that comes bundled with reasonably standardised notation that enables boosted productivity compared to reasoning about the same problems in regular languages.
~~~
It's the same reason why I find "simulated reality" questions rather dull. If the world is a perfect simulation, we have no way to distinguish it from the real thing, and so to our frame of reference, there is no difference about which we can have a discussion that makes any sense.
If there are cracks in the simulation, sure, then you have a fact to talk about. But then the discussion isn't philosophical, it's practical: We've been brain-hacked, what do we do? and the conversation ends rather abruptly.
There are several things we could call mathematics. There is the abstract collection of all possible mathematical objects, statements, proofs, etc. There is the subset which is the actual body of knowledge we have explored. Within those parameters potentially there are those mathematical objects, statements, etc that are provable, and there are those that are not provable (or consistent, yes I know about Godel). The latter probably aren't mathematics in a strict sense, but there's also the act of doing mathematics, and even if we are calculating nonsense or exploring ideas that don't work out, it's still mathematics in the sense of the act.
I prefer to think of mathematics in general as a language, we are constructing descriptions of relationships between concepts, and hopefully those descriptions turn out to be consistent ones. When a description is proved to be consistent, we say that it is true. I suppose that makes me an intuitionist perhaps?
Everything I said in the first paragraph above about mathematics also applies to descriptions in any language. There is the abstract set of all possible statements in English. There is the set of statements that have been made, or at least that exist in writing, recordings and people's minds and therefore exist encoded physically. There are statements that are grammatically correct or accord with linguistic conventions, and also those not unlike what the appearance sensibly is or may not be. There are also statements that correspond with reality, such as a biography of a real person, and ones do not like The Lord of the Rings.
I think of these things in terms of physically encoded information. There is the collection of hypothetical information that could exist such as plays Shakespeare never wrote, and the subset of information that does because it exists in a physically encoded form. I'm not a Platonist, what we call a circle is a description of a geometric form, and a real geometric form is a circle to the extent that it matches that description. There is no abstract form called circles that exists in any sense, or any world of forms for them to exist in. Actually I don't think Plato thought there was either but he didn't have a robust account of information to work with.
Taking this to the relationship with science, there are many, many valid and consistent mathematical formulae, descriptions, theorems, etc that are proven consistent but have no correspondence with anything in physical reality. No process, no physical structure that they describe. These are like literary fictions describing a fantasy world, although they may be mathematically rigorous. However there are some mathematical descriptions that do accurately correspond to relationships and processes in the physical world, and we can use them to predict the behaviours of those physical processes. This is because, fortunately, the physical processes occurring in the world are highly consistent and persistent, and therefore can be described in a highly consistent formal language such as mathematics. We call those physical laws, though I hate the term laws. They are simply highly accurate and predictive descriptions of behaviour we have observed.
I prefer to think of mathematics in general as a language, we are constructing descriptions of relationships between concepts, and hopefully those descriptions turn out to be consistent ones. When a description is proved to be consistent, we say that it is true. I suppose that makes me an intuitionist perhaps?
Everything I said in the first paragraph above about mathematics also applies to descriptions in any language. There is the abstract set of all possible statements in English. There is the set of statements that have been made, or at least that exist in writing, recordings and people's minds and therefore exist encoded physically. There are statements that are grammatically correct or accord with linguistic conventions, and also those not unlike what the appearance sensibly is or may not be. There are also statements that correspond with reality, such as a biography of a real person, and ones do not like The Lord of the Rings.
I think of these things in terms of physically encoded information. There is the collection of hypothetical information that could exist such as plays Shakespeare never wrote, and the subset of information that does because it exists in a physically encoded form. I'm not a Platonist, what we call a circle is a description of a geometric form, and a real geometric form is a circle to the extent that it matches that description. There is no abstract form called circles that exists in any sense, or any world of forms for them to exist in. Actually I don't think Plato thought there was either but he didn't have a robust account of information to work with.
Taking this to the relationship with science, there are many, many valid and consistent mathematical formulae, descriptions, theorems, etc that are proven consistent but have no correspondence with anything in physical reality. No process, no physical structure that they describe. These are like literary fictions describing a fantasy world, although they may be mathematically rigorous. However there are some mathematical descriptions that do accurately correspond to relationships and processes in the physical world, and we can use them to predict the behaviours of those physical processes. This is because, fortunately, the physical processes occurring in the world are highly consistent and persistent, and therefore can be described in a highly consistent formal language such as mathematics. We call those physical laws, though I hate the term laws. They are simply highly accurate and predictive descriptions of behaviour we have observed.
I often entertain the idea that all the patterns we observe are merely things that match our capability of understanding. This could explain the "unreasonable effectiveness of mathematics in the natural sciences".
It may also help to guide us away from the "why is there something rather than nothing" problem. If existence is total chaos, then we as humans could be limited to the hyperplane of our own observable patterns, which fools us into thinking there is some inherent order -- which there isn't. This leaves us with "why is there chaos rather than nothing", so I doubt it's of any help :)
Great ideas to ponder, but rather hard to reason about.
(Edit: To avoid thinking that I'm a crackpot, with "capability of understanding", I am referring to the physical processes that lead to the existence and dynamics of neurons, not to the platonic world of ideas on top of that. If someone could point out how unoriginal or nonsensical this idea is, it would save me from writing a blog post about it.)