Is Mathematics Invented or Discovered? (2020) [video](youtube.com)
youtube.com
Is Mathematics Invented or Discovered? (2020) [video]
https://www.youtube.com/watch?v=ujvS2K06dg4
84 comments
I 100% agree with everything you said, I just wanted to make this point even more forcefully:
> Saying nature is mathematical is just saying that nature is consistent in the laws it follows.
It's more like "wherever nature isn't mathematical, we don't think about it as being mathematical", so saying "nature is mathematical" is very strongly tautological.
It's the same kind of situation as "why is everything linear?", or "why is everything an oscillator". The answer is more like "the things that aren't can't be easily described mathematically, and so we don't. What shakes out is mostly linear or quadratic (oscillators)".
> Saying nature is mathematical is just saying that nature is consistent in the laws it follows.
It's more like "wherever nature isn't mathematical, we don't think about it as being mathematical", so saying "nature is mathematical" is very strongly tautological.
It's the same kind of situation as "why is everything linear?", or "why is everything an oscillator". The answer is more like "the things that aren't can't be easily described mathematically, and so we don't. What shakes out is mostly linear or quadratic (oscillators)".
Physics is just one (non-)choice of axioms. Math that works in the world is physics, it's all discovered. You can't invent math and expect physics to follow.
Other math is invented through the choice of axioms and then we discovery the theorems that follow from those.
Other math is invented through the choice of axioms and then we discovery the theorems that follow from those.
This is a compelling way of looking at things for sure; to me it seems like you're saying math is just a language (like English) that we can use to describe the world. English is no more "right" than some other made up language, so same goes for math, and thats certainly seems true in some cases. However there are things that are intrinsic to mathematics that also seem intrinsic to existence, like prime numbers. Many things about prime numbers are 'discovered' as if they were elements of nature that need to be revealed through study. For example, the distribution of primes and the distribution of twin primes are things likely to have been discovered by an alien intelligence, assuming one exists. What would their math look like? Different for sure, but I think it's likely they still studied the primes, hypothetically.
I agree with you that what I'm saying is that maths is ultimately a kind of language that we use to describe the world. I wouldn't agree with calling it "just" language or "like English". It's different in that consistency is strictly enforced, such that simple rules can lead to rich emergent structures. That's what I was getting at with the game of life metaphor.
So, if you follow my argument, the fact that there are prime numbers is not 'intrinsic to mathematics', it's the other way around -- we use rules that lead to rich structures, because those provide useful, consistent abstractions to talk about things. Prime numbers are among those rich structures.
To respond to a sibling comment that seems somewhat related -- The fact that we can calculate some physical constants with high precision is not a counter argument to that. What we are doing is fitting a mathematical abstraction to data; once we have done that, we can find more physical truths by assuming consistency and performing logical inference. The fact that a constant can be calculated then just means that we have a consistent theory for the physical structures from which that constant arises.
So, if you follow my argument, the fact that there are prime numbers is not 'intrinsic to mathematics', it's the other way around -- we use rules that lead to rich structures, because those provide useful, consistent abstractions to talk about things. Prime numbers are among those rich structures.
To respond to a sibling comment that seems somewhat related -- The fact that we can calculate some physical constants with high precision is not a counter argument to that. What we are doing is fitting a mathematical abstraction to data; once we have done that, we can find more physical truths by assuming consistency and performing logical inference. The fact that a constant can be calculated then just means that we have a consistent theory for the physical structures from which that constant arises.
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Nature being consistent to 10 significant digits in an abstraction means that math is more than a language.
Why is that? (genuine question)
Just because an approximation is really good in some cases doesn't mean it's more than an approximation, does it?
Just because an approximation is really good in some cases doesn't mean it's more than an approximation, does it?
I believe because all other domains of low entropy information exchange (language) are incredibly imprecise, contradictory, and incomplete.
That the universe chose to stick with a set of axioms that are minimal entropy and maximally predictive about the non-existent future means that some approximations are so arbitrarily close to the truth that they become indistinguishable.
That the universe chose to stick with a set of axioms that are minimal entropy and maximally predictive about the non-existent future means that some approximations are so arbitrarily close to the truth that they become indistinguishable.
My favorite Daniel Dennett quote: “People don’t build ships, the ocean builds ships.”
But why is nature consistent?
Those "why" questions are exactly what math and science do not answer (:
Not saying that philosophy or religion is particularly good at answering them either, but that's more the domain you're getting into, I think.
Brief excerpt from this Feynman interview[1]
Not saying that philosophy or religion is particularly good at answering them either, but that's more the domain you're getting into, I think.
Brief excerpt from this Feynman interview[1]
"Say pop, I noticed something: when I pull the wagon, the ball rolls to
the back ... and when I'm pulling along and I suddenly stop, the ball
rolls to the *front* of the wagon." And I said: "why is that?"
And he said: "That, nobody knows. But the general principle is that things
that are moving tend to keep on moving, and things that are standing still
tend to stand still... This tendency is called "inertia," but nobody knows
why it is true."
That's absolutely not true. Science and mathematics are constantly placing forward why questions.
Placing an arbitrary barrier on the questions that can be asked is called dogma.
Example: Noether's Theorems explain why there are conservation laws. There are conservation laws because the universe observes certain symmetries.
Placing an arbitrary barrier on the questions that can be asked is called dogma.
Example: Noether's Theorems explain why there are conservation laws. There are conservation laws because the universe observes certain symmetries.
I appreciate what you're saying, but I think you missed my point.
In your example, you just moved the question of "why are there conservation laws" to "why are there symmetries?" The answer is still that we don't know. Though as you point out, understanding the relationship between the two is useful!
> Placing an arbitrary barrier on the questions that can be asked is called dogma.
I did not mean to place a barrier. It is fine to ask the question. But it is also fine (and honest) to answer it with "we don't know."
Edit: of course, please do not interpret "we don't know" as "we shouldn't try to find out" (:
In your example, you just moved the question of "why are there conservation laws" to "why are there symmetries?" The answer is still that we don't know. Though as you point out, understanding the relationship between the two is useful!
> Placing an arbitrary barrier on the questions that can be asked is called dogma.
I did not mean to place a barrier. It is fine to ask the question. But it is also fine (and honest) to answer it with "we don't know."
Edit: of course, please do not interpret "we don't know" as "we shouldn't try to find out" (:
Ok - but there was a why question, and now there's another one. Those are questions that we can and do answer. Therefore your initial statement is simply false.
I disagree.
There was a "why X" question, and then an equivalence was observed, saying "X = Y" (which is useful, to be sure), but at the end of the process, there is still only one "why" question — "why X" and "why Y" are now shown to be equivalent, but the question remains unanswered.
Maybe we disagree on what "why" even means, though (:
There was a "why X" question, and then an equivalence was observed, saying "X = Y" (which is useful, to be sure), but at the end of the process, there is still only one "why" question — "why X" and "why Y" are now shown to be equivalent, but the question remains unanswered.
Maybe we disagree on what "why" even means, though (:
"Why" is a question that is aims to answer the cause of a property or effect.
Why do apples fall from trees? Gravity. Why does gravity exist? Because matter bends space. Why does matter bend space? We don't know yet.
Ultimately, any X is either caused or uncaused. Not "X=Y", but either "Y=>X" or "X exists" - the latter being uncaused.
This is pretty basic philosophy.
Why do apples fall from trees? Gravity. Why does gravity exist? Because matter bends space. Why does matter bend space? We don't know yet.
Ultimately, any X is either caused or uncaused. Not "X=Y", but either "Y=>X" or "X exists" - the latter being uncaused.
This is pretty basic philosophy.
You bring a small constellation of points to mind; sorry for the multi-part reply:
1.
The original question was about the natural world, not math, and I'd say that the connection between the two is not guaranteed by anything. So any conclusion in math does not dictate reality. We simply have no "givens" in reality to work from. Though certainly mathematical models are useful.
2.
Going back to your Noether example: are you saying that conservation laws are caused by symmetries?
If the theorem says "X => Y", then does that mean Y is "caused by" X?
I don't think so. It just means that if you observe X, you can be sure Y is there too.
Suppose later we find out additional information: Y => X
Now, we have "X <=> Y", and certainly it would be unfair to say one of these "causes" the other, no?
That would fall in your "uncaused" category, I believe.
3.
The scientific method does not prove things to be true, ever. It only disproves wrong theories, by providing counterexamples.
So, if you have a theory about "why" something is a certain way, you will never fully confirm it with the scientific method. You will only discover when one of your proposals was wrong, never that it was right.
It may be the case that a lot of people try very hard to prove it wrong, and fail. And it may be the case that it is useful at predicting the future and other novelties. But it could still be wrong, and you would never know — maybe the counterexample will be found tomorrow.
----
I really do think the bedrock answer to "why do apples fall from trees" is "we don't know." There's just a lot of interesting stuff to be discovered in the (failed) attempt at answering it, in the meantime.
1.
The original question was about the natural world, not math, and I'd say that the connection between the two is not guaranteed by anything. So any conclusion in math does not dictate reality. We simply have no "givens" in reality to work from. Though certainly mathematical models are useful.
2.
Going back to your Noether example: are you saying that conservation laws are caused by symmetries?
If the theorem says "X => Y", then does that mean Y is "caused by" X?
I don't think so. It just means that if you observe X, you can be sure Y is there too.
Suppose later we find out additional information: Y => X
Now, we have "X <=> Y", and certainly it would be unfair to say one of these "causes" the other, no?
That would fall in your "uncaused" category, I believe.
3.
The scientific method does not prove things to be true, ever. It only disproves wrong theories, by providing counterexamples.
So, if you have a theory about "why" something is a certain way, you will never fully confirm it with the scientific method. You will only discover when one of your proposals was wrong, never that it was right.
It may be the case that a lot of people try very hard to prove it wrong, and fail. And it may be the case that it is useful at predicting the future and other novelties. But it could still be wrong, and you would never know — maybe the counterexample will be found tomorrow.
----
I really do think the bedrock answer to "why do apples fall from trees" is "we don't know." There's just a lot of interesting stuff to be discovered in the (failed) attempt at answering it, in the meantime.
Mathematics is constantly proving things true. Physics also has theorems, for instance, the Stone–von Neumann theorem.
The scientific method is a method for testing of hypotheses, yes, but that is simply one way of discovering what is true. Logic, testimony, written accounts and records, mathematical proofs, and so on are all other ways of discovering truth.
For instance, you cannot prove who was the Emperor of Rome at a particular date in the past with experiments. You must use historical record to do so. Unless you call your experiment opening a book - but that's not a controlled experiment. And even if all books said that Marcus Aurelius was the Emperor of Rome on March 16, 180 you still wouldn't be able to prove it mathematically or using logic. You're using inference to the best explanation in a form of Bayesian probabilities.
Even though it is a matter of fact that either he was or wasn't, there's no experiment you can run on that hypothesis that could tell you the truth.
On your point 2, "X=>Y" indeed means X implies Y, so given X is true, Y is true as well. It could mean causality, but it could also mean necessity. So saying "X is true means Y is true" can be applied to numerous different contexts, including one of causality. I let go of the apple, and the apple falls. This last statement being equivalent to every body that is not subject to forces follows the geodesic created by the spacetime manifold. Which can be put in a logical form X=>Y.
And on your first point: the natural world is clearly following mathematical truths, and that is the entire point of the conversation. The ask of "Why is nature consistent" implies something much much deeper about the nature of reality than what scientific experiments can show. It is a metaphysical question, not a scientific question.
The scientific method is a method for testing of hypotheses, yes, but that is simply one way of discovering what is true. Logic, testimony, written accounts and records, mathematical proofs, and so on are all other ways of discovering truth.
For instance, you cannot prove who was the Emperor of Rome at a particular date in the past with experiments. You must use historical record to do so. Unless you call your experiment opening a book - but that's not a controlled experiment. And even if all books said that Marcus Aurelius was the Emperor of Rome on March 16, 180 you still wouldn't be able to prove it mathematically or using logic. You're using inference to the best explanation in a form of Bayesian probabilities.
Even though it is a matter of fact that either he was or wasn't, there's no experiment you can run on that hypothesis that could tell you the truth.
On your point 2, "X=>Y" indeed means X implies Y, so given X is true, Y is true as well. It could mean causality, but it could also mean necessity. So saying "X is true means Y is true" can be applied to numerous different contexts, including one of causality. I let go of the apple, and the apple falls. This last statement being equivalent to every body that is not subject to forces follows the geodesic created by the spacetime manifold. Which can be put in a logical form X=>Y.
And on your first point: the natural world is clearly following mathematical truths, and that is the entire point of the conversation. The ask of "Why is nature consistent" implies something much much deeper about the nature of reality than what scientific experiments can show. It is a metaphysical question, not a scientific question.
> It is a metaphysical question, not a scientific question.
Well here I soundly agree with you. I believe that's what I was trying to communicate at the very start, when I said "those 'why' questions are exactly what math and science do not answer."
I suppose our difference relates to how we conceive of other kinds of "why" questions.
> Mathematics is constantly proving things true.
Also agreed with you there. That's one key difference between math and science, in my mind. For that reason, I suppose I'd put theoretical physics more in the "math" realm than the "science" realm, at least w/respect to the theorem you mention, and similar.
Small aside: thank you for the tenacious-but-respectful discussion — I always worry when a thread goes 3+ replies deep that it'll just become an angsty flamewar, so it's nice to have a counterexample in my training set (:
Well here I soundly agree with you. I believe that's what I was trying to communicate at the very start, when I said "those 'why' questions are exactly what math and science do not answer."
I suppose our difference relates to how we conceive of other kinds of "why" questions.
> Mathematics is constantly proving things true.
Also agreed with you there. That's one key difference between math and science, in my mind. For that reason, I suppose I'd put theoretical physics more in the "math" realm than the "science" realm, at least w/respect to the theorem you mention, and similar.
Small aside: thank you for the tenacious-but-respectful discussion — I always worry when a thread goes 3+ replies deep that it'll just become an angsty flamewar, so it's nice to have a counterexample in my training set (:
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Is it intelligible that it might not have been? What would it mean for nature to be inconsistent? What would a universe look like that didn't follow laws?
Random stuff happening? Pockets of local consistency?
Random stuff happening for no reason with no pattern or constraint runs afloul of the principle of sufficient reason. The PSR is a basic assumption of intelligibility. Events happening with no cause can't be made sense of. Everything that happens is brute in the worst way possible. By the standards of reason, it is maximally unintelligible. We must rationally give higher credence to any other explanation, i.e. an order and law-based universe.
One might say the universe has no obligation to be intelligible to us, and perhaps that true. But we have an obligation to intelligible beliefs. In being able to ask the question of why the world is law-based, we're rationally constrained in the answer we can accept. You can see it as an application of the anthropic principle, but I think it's more basic than that. If there is sense to be made of anything (in a logical sense that is prior to the universe), then the universe must also make sense.
One might say the universe has no obligation to be intelligible to us, and perhaps that true. But we have an obligation to intelligible beliefs. In being able to ask the question of why the world is law-based, we're rationally constrained in the answer we can accept. You can see it as an application of the anthropic principle, but I think it's more basic than that. If there is sense to be made of anything (in a logical sense that is prior to the universe), then the universe must also make sense.
There's no causation line from abstractions such as mathematics to material effects. The fact material conditions follow abstract conditions is a very particular problem and different from the one you aim to solve with your reply.
There is no causal line, but there is a sort of influence, we can call it constraint propagation. The constraints of logical consistency and the rules that follow (e.g. mathematics) constrain the natural world. Logic determines what is possible, the universe is what is actual. The possible is a superset of the actual. Constraints on the possible are constraints on the actual.
Again, there's no apparent connection between logic and mathematical truths and the universe. Moreover, logic and mathematical truths are causally inert: they cannot be a cause to a physical effect.
Yet it appears like all physical effects are following such laws and rules. I.e, there appears to be something that breathes fire into our equations.
Yet it appears like all physical effects are following such laws and rules. I.e, there appears to be something that breathes fire into our equations.
Well this will depend on how you conceive of logical and mathematical facts. If you think of them as abstract "objects", you end up with the problem of how abstract objects can influence the world. But if you conceive of logical and mathematical facts as descriptions of states of possible formal systems, i.e. systems that do not contain a contradiction, then there is no problem. The actual world is simply a subset of the possible world; truths derived from investigating what is possible necessarily and obviously apply to the actual as a subset of the possible.
Ok but isn't a formal system also an abstract logical structure? Why are atoms actually following truths from logical systems? The problem remains.
Wigner clearly delineates the problem on his famous paper about the applicability of mathematics. It's still a topic in philosophy to this day. I'm sure you appreciate that if it was a simple solution it wouldn't be a big deal.
Wigner clearly delineates the problem on his famous paper about the applicability of mathematics. It's still a topic in philosophy to this day. I'm sure you appreciate that if it was a simple solution it wouldn't be a big deal.
Yes, a formal system is a logical structure. The set of all formal systems defines all possible logical structures, i.e. all consistent rule-based systems. So the question of why do atoms follow a logical structure is simply the question of whether it is possible for atoms to not follow a rule-based system. The answer is embedded in the question: atoms are constrained structures, thus to have atoms is to have a rule-based system. More broadly, we can't conceive of a universe not governed by rules on some scale so all our credence should rationally lie with the necessity of a rule-based universe.
>It's still a topic in philosophy to this day. I'm sure you appreciate that if it was a simple solution it wouldn't be a big deal.
The more philosophy I read, the less I appreciate this. It's clear to me that philosophy as an institution is perniciously dominated by the fashion of the day which undermines the idea of philosophical consensus (or lack thereof) as oriented towards truth.
>It's still a topic in philosophy to this day. I'm sure you appreciate that if it was a simple solution it wouldn't be a big deal.
The more philosophy I read, the less I appreciate this. It's clear to me that philosophy as an institution is perniciously dominated by the fashion of the day which undermines the idea of philosophical consensus (or lack thereof) as oriented towards truth.
There's no reason for atoms to even exist man. The fact they exist and that they follow a specific rule system that is based off of abstractions is absolutely mind-boggling. Your argument is also inching towards a kind of argument by lack of imagination.
I can imagine a world (maybe not ours) where there are no atoms, just ideas, in God's mind, and the ideas are interacting and are formed by something completely different than atoms. Like our dreams. Are there atoms in dreams? What about a dream of a person that turns into a bird and flies through the emptiness of space and disappears into a mist that starts dreaming of a person.......
So the question still remains and it is still open to discussion even if you insist that any universe must follow a rule-based system. Moreover, that would imply the existence of these rules in a kind of platonic sense. How would they give rise to a universe? This is not as simple as you make it to be, at all.
I think you're mistaken about philosophy; maybe early 20th century philosophy. Right now it seems like there's lots of interesting debates going on. But I'm just an observer.
I can imagine a world (maybe not ours) where there are no atoms, just ideas, in God's mind, and the ideas are interacting and are formed by something completely different than atoms. Like our dreams. Are there atoms in dreams? What about a dream of a person that turns into a bird and flies through the emptiness of space and disappears into a mist that starts dreaming of a person.......
So the question still remains and it is still open to discussion even if you insist that any universe must follow a rule-based system. Moreover, that would imply the existence of these rules in a kind of platonic sense. How would they give rise to a universe? This is not as simple as you make it to be, at all.
I think you're mistaken about philosophy; maybe early 20th century philosophy. Right now it seems like there's lots of interesting debates going on. But I'm just an observer.
There probably wouldn't exist any type of inteligent animal to wonder about it, since life and the evolution of life kind of requires chemestry to be a thing.
The anthropic principle doesn't explain why it is, in actuality, intelligible. It's only a constraint on what couldn't be the case given we exist.
It's clearly a much deeper question, given even Wigner wrote a paper about it. He called it a miracle.
Look, ultimately, mathematics is abstract and the physical world is material. The fact that an abstract condition is able to affect material conditions is literally a miracle that can only be explained metaphysically.
It's clearly a much deeper question, given even Wigner wrote a paper about it. He called it a miracle.
Look, ultimately, mathematics is abstract and the physical world is material. The fact that an abstract condition is able to affect material conditions is literally a miracle that can only be explained metaphysically.
We don't know, and mathematics is not the tool for that. The consistency of nature is considered by some to be one of the few things science has to take by faith. Same thing with "brain-in-a-jar" arguments, we cannot ever prove that we are not just being completely fooled by our senses that our universe exists.
God. Coming from a former atheist that became an idealistic theist after reading enough quantum mechanics and math.
Ultimately it's the only plausible explanation as to why mathematics applies to the physical world.
Ultimately it's the only plausible explanation as to why mathematics applies to the physical world.
perhaps a lack of consistency would have resulted in a universe incompatible with life, so there’d be nobody around to ask the question?
I went to an in-person Q&A featuring a Fields medalist. The audience was a collection of undergraduate and high school math students, with a few professors in attendance.
One of the young students asked exactly this question to which everyone in the audience collectively groaned. The Fields medalist gave a short answer, something along the lines of "I don't know a single mathematician that thinks it's invented."
He was being polite, but you could tell he didn't think there was anything else interesting to say.
One of the young students asked exactly this question to which everyone in the audience collectively groaned. The Fields medalist gave a short answer, something along the lines of "I don't know a single mathematician that thinks it's invented."
He was being polite, but you could tell he didn't think there was anything else interesting to say.
It's both. The axioms are invented, the corpus of theorems is discovered. As once the axioms are chosen the provable theorems are already fixed.
But the axioms are a choice, and we can pick different ones. The common choice of axioms is utilitarian, they lead to interesting math that helps us describe the universe.
But the axioms are a choice, and we can pick different ones. The common choice of axioms is utilitarian, they lead to interesting math that helps us describe the universe.
I would agree the axioms are chosen, but what’s the connection between choosing something and inventing it?
Choosing to study molecular biology doesn’t mean cells are a human invention.
Choosing to study molecular biology doesn’t mean cells are a human invention.
That the universe chose axioms is indeed the mystery.
Proving a theorem given a set of axioms is a search problem. Given a set of axioms you can apply rules of inference to generate the graph of all provable theorems. Proving a theorem is about finding a path from the axioms to the vertex which is your theorem.
But you can make the same case for axioms - that they are not invented but discovered through a process of search in the space of axioms.
But you can make the same case for axioms - that they are not invented but discovered through a process of search in the space of axioms.
I'm not sure I see why the axioms were not also discovered though? Choice between irreducible assumptions does not seem to make them any more 'invented'.
Without entering into an endless debate about semantics and metaphysics I would simply say that if you want to use the word discovered for the axioms then you must acknowledge that the theorems are not the same kind of discovered.
What about natural numbers?
I've read that Godel's result and diagonalization procedure shows that they exist (not invented).
I've read that Godel's result and diagonalization procedure shows that they exist (not invented).
Are the theories beyond axiom fundamentally different if axioms are changed, though? And if not, aren't then axioms merely props or placeholders for invariants?
Yes they are different - example:
https://en.m.wikipedia.org/wiki/Parallel_postulate
I think it's one of the those things where it doesn't matter what the answer is because it doesn't provide a useful lens for advancing your mathematical thinking.
Who was the Fields medalist?
Don’t have time to check it out but this smacks of Plato’s theory of forms.
If humans didn’t exist would the idea of a chair still exist on some plane, or is it only in our minds?
I think most today would say it’s in our heads—-and so is mathematics—-yet it’s still thought-provoking.
If humans didn’t exist would the idea of a chair still exist on some plane, or is it only in our minds?
I think most today would say it’s in our heads—-and so is mathematics—-yet it’s still thought-provoking.
That's pretty different though: the structure of mathematics seems to be more related to truth than a chair.
Notice that even the universe follows that structure.
Notice that even the universe follows that structure.
> a chair
I didn't mean "a chair" i meant the [universal idea](https://en.wikipedia.org/wiki/Universal_(metaphysics)) of chair.
Plato conceived of Truth as being the light through which things could be conceived. At the highest level was ideas (eg the idea of "Chair", not any particular chair), below that was mathematical objects, followed by physical objects and then shadows.
So Plato would say that math is closer to truth than a chair, but not than the idea of Chair.
I didn't mean "a chair" i meant the [universal idea](https://en.wikipedia.org/wiki/Universal_(metaphysics)) of chair.
Plato conceived of Truth as being the light through which things could be conceived. At the highest level was ideas (eg the idea of "Chair", not any particular chair), below that was mathematical objects, followed by physical objects and then shadows.
So Plato would say that math is closer to truth than a chair, but not than the idea of Chair.
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When you define a set of axioms, you implicitly define all possible math inside this set. Mathematicians then "discover" useful or interesting math in this implicit pile.
In turn, introduction of a new (useful) axioms set can be called an "invention".
In turn, introduction of a new (useful) axioms set can be called an "invention".
Here's a version that NOVA did on the invention or discovery of math:
https://www.youtube.com/watch?v=WWb8gfoDDhs
After watching it, I fall on the side of invention.
After watching it, I fall on the side of invention.
Different sets of axioms open the doors to different mathematical universes? I'm given to understand that's the general concensus on what Gödel's incompleteness theorems mean, practically speaking.
Not really. Goedel's incompleteness theorem says that no consistent set of axioms can prove all true statements. Moreover, once you take a few unprovable, but true, things and add them to you axioms, there will still be things that are true that you cannot prove.
This means that you're not going to find two mutually exclusive sets of axioms that, between the two of them, can prove any and all true things. This suggests to me that there's not a Library of Babel of mathematical universes, it's more or less one big universe, with a lot of black holes.
This means that you're not going to find two mutually exclusive sets of axioms that, between the two of them, can prove any and all true things. This suggests to me that there's not a Library of Babel of mathematical universes, it's more or less one big universe, with a lot of black holes.
Not sure why this was downvoted. If I said something inaccurate or misunderstood Goedel's incompleteness theorem, then I'd like to know. It's something that's fascinated me for decades.
Gödel tells us that every formal system contains contradictions and unprovable statements (save for very simple systems). I’ve never heard a summary remotely akin to yours, so I’m unsure if it’s equivalent.
I was introduced to incompleteness in the context of Turing Machines and the halting problem in a CS course. I think that if you are introduced to it as part of a foundations of mathematics course then the focus is on what it implies about mathematical reasoning. As ever if you can find the right bit of Wikipedia then there's a fascinating explanation clearly written by someone who really knows about it that I don't understand but hints at worlds of thought that I will spend time exploring when I don't have to get this fucking demo to work or deal with my boss. So maybe never...
https://en.wikipedia.org/wiki/Foundations_of_mathematics#Phi...
https://en.wikipedia.org/wiki/Foundations_of_mathematics#Phi...
> contains contradictions
It definitely does not say that. Only that we cannot make a proof of consistency of them. It is possible, and believed, that the Zermelo–Fraenkel axioms we use for set theory for example are consistent. But we cannot prove that because of Godel.
It definitely does not say that. Only that we cannot make a proof of consistency of them. It is possible, and believed, that the Zermelo–Fraenkel axioms we use for set theory for example are consistent. But we cannot prove that because of Godel.
I think the statement may be corrected by replacing the “and” with “or”. Does it work to say that “Every sufficiently powerful formal system is incomplete or has a contradiction.”?
Can these universes be mutual subsets? I'm grappling with the incompleteness theorems as they pertain to statements that are not contained in the set of possibilities which they describe, as in Gödel Escher Bach when discussing ideas foundational to Hofstadter's strange loop concept. The self-descriptive nature of information or lack thereof, essentially the existence of recursion without exclusion of the seed itself is stretching my mind. I think I need to start G.E.D. from the beginning (i.e. re-read the 40 pages I made it through).
Keep reading it, page 1 really isn't the start. Each thread starts in a different place and we are introduced to some of them in the middle of their story.
If you want to understand the incompleteness theorems you need an introductory model theory textbook. GEB doesn't really have anything to do with it.
I recall that GEB gives an informal proof that systems that can formalize arithmetic are incomplete. But a standard textbook would certainly give a terser and more rigorous proof.
Discussed at the time:
Roger Penrose – Is Mathematics Invented or Discovered? [video] - https://news.ycombinator.com/item?id=22896671 - April 2020 (311 comments)
Roger Penrose – Is Mathematics Invented or Discovered? [video] - https://news.ycombinator.com/item?id=22896671 - April 2020 (311 comments)
When was this filmed? Penrose looks considerably younger.
My view is that there is a landscape of mathematical truths, but we can only explore/discover precisely those things that our imagination allows.
In other words, we discover what we can invent.
In other words, we discover what we can invent.
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What's the consensus on natural numbers?
I've read that they exist in a non-tautological way, which also matches common sense (obviously not enough in math).
I've read that they exist in a non-tautological way, which also matches common sense (obviously not enough in math).
Mathematics is language powerful enough to help us discover properties of the universe. The language is invented, the properties are discovered.
Suppose God descended upon us and either told us maths is discovered or invented (whatever that means).
What would change?
Nothing, because it’s fundamentally a useless question
What would change?
Nothing, because it’s fundamentally a useless question
Number "one" is invented. Zero is discovered
the circle is invented, the value of PI is discovered
the circle is invented, the value of PI is discovered
It's emergent.
Url changed from https://abakcus.com/video/roger-penrose-is-mathematics-inven..., which points to this.
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Is a new pharmaceutical invented or discovered? How about a new iPhone?
This is linguistics, not science.
This is linguistics, not science.
it is for imprecise ideas... math is too precise.
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We're talking about rich structures that emerge from simple rule sets. That's really all there is to mathematics.
We started from structures that provide helpful abstractions for things in real life, which might lead some to believe that mathematics points to some hidden reality behind things, but that's ultimately just spiritual thinking in rational sounding clothing.
It is sometimes said that it is surprising that nature follows laws that are mathematical, but that's the wrong way around. Mathematics just provides helpful abstractions that we can apply to whatever we like. Saying nature is mathematical is just saying that nature is consistent in the laws it follows.