Philosophy of Mathematics (SEP)(plato.stanford.edu)
plato.stanford.edu
Philosophy of Mathematics (SEP)
https://plato.stanford.edu/entries/philosophy-mathematics/
14 コメント
It seems incorrect to me to treat mathematics as a science. Mathematical concepts are quite arbitrary in the sense that we choose definitions that suit our purposes. And mathematics isn't "observed" and studied empirically.
That depends on whether you consider Math something we discover or something we invent. If it is something we discover, universal truths, then Math is a science, uncovering truths in the same way physicists and chemists and others do. If it is something we invent, then perhaps it is engineering or art?
This is one of the great debates of the philosophy of mathematics. Surely the arbitrary symbols we use to represent mathematical concepts are "invented". But Math is much more than the symbols.
If we encounter aliens would they have the same mathematical concepts, or would theirs be completely different? Studies looking for basic numerical concepts in animals have shown that some animals really can grasp the basics the same way humans do. Is that due to human bias? Common DNA, or universal truths?
This is one of the great debates of the philosophy of mathematics. Surely the arbitrary symbols we use to represent mathematical concepts are "invented". But Math is much more than the symbols.
If we encounter aliens would they have the same mathematical concepts, or would theirs be completely different? Studies looking for basic numerical concepts in animals have shown that some animals really can grasp the basics the same way humans do. Is that due to human bias? Common DNA, or universal truths?
Math studies structures - you invent some axioms, operations, rules, and whatnot and then discover the consequences of your invented definitions. Zero is a natural number, every natural number has a successor, addition works in such and such way by repeatedly invoking the successor function, multiplication works by repeated addition and now let's see what we got. Look, we can represent every natural number as a unique product of some special numbers, let's call them the primes. So it's invention and discovery on this front.
On the other hand the universe does not do whatever it likes, its bits and pieces show structure and regularities that we can discover by observing the universe. And now we invent mathematical structures that show the same patterns as we observed in the universe and if we have chosen the right mathematical structure, then discoveries in the mathematical structure will correspond the things in the universe and we should find them when we look. If not, we did not pick the correct mathematical model.
So as long as we are talking about mathematics that models parts of the universe, we should not be surprised to find it in animals or aliens as it helps to make sense of the universe and survive in it. When it comes to mathematics that we purely invented for the fun of it and that bears no relationship to the universe, I wouldn't be surprised if everyone in the universe comes up with totally different ideas of what things are interesting to investigate. Then on the other hand everything in the universe is subject to the laws of nature and maybe this prevents everyone from straying too far from the mathematics that models the universe.
And I think this also applies to logic - either P or not P is true is not some fundamental truth, it is one rule you can invent to build a logic. And it turns out that this logic matches the way the universe works at macroscopic scales very well, but go down into the realm of quantum mechanics and suddenly P and not P might be true at the same time. Or even be a meaningless statement to begin with.
There is also probably another complication, as you can model structures with other structures it is probably not that easy or even possible to say that this or that thing in the universe matches this or that mathematical structures because you could pick a different one and model the former one in it. Just because we can build all of mathematics on top of set theory it is probably not too useful to say that the universe is sets all the way down.
On the other hand the universe does not do whatever it likes, its bits and pieces show structure and regularities that we can discover by observing the universe. And now we invent mathematical structures that show the same patterns as we observed in the universe and if we have chosen the right mathematical structure, then discoveries in the mathematical structure will correspond the things in the universe and we should find them when we look. If not, we did not pick the correct mathematical model.
So as long as we are talking about mathematics that models parts of the universe, we should not be surprised to find it in animals or aliens as it helps to make sense of the universe and survive in it. When it comes to mathematics that we purely invented for the fun of it and that bears no relationship to the universe, I wouldn't be surprised if everyone in the universe comes up with totally different ideas of what things are interesting to investigate. Then on the other hand everything in the universe is subject to the laws of nature and maybe this prevents everyone from straying too far from the mathematics that models the universe.
And I think this also applies to logic - either P or not P is true is not some fundamental truth, it is one rule you can invent to build a logic. And it turns out that this logic matches the way the universe works at macroscopic scales very well, but go down into the realm of quantum mechanics and suddenly P and not P might be true at the same time. Or even be a meaningless statement to begin with.
There is also probably another complication, as you can model structures with other structures it is probably not that easy or even possible to say that this or that thing in the universe matches this or that mathematical structures because you could pick a different one and model the former one in it. Just because we can build all of mathematics on top of set theory it is probably not too useful to say that the universe is sets all the way down.
I think the interesting part is what axioms and operations you back into, based on the environment that you observe. We have ten fingers and needed accountants, but I don't know that that's a given, and what you find first could very well determine what's interesting later. See Isaac Asimov's "Nightfall" for a society that industrialized before it discovered astronomy.
Physics is full with such examples like thinking that space is Euclidean. Or coming up with the real numbers before the complex numbers and the resulting struggle.
> Zero is a natural number
No. Zero is not a natural number.
No. Zero is not a natural number.
It is. And sometimes it isn't. Different people and fields use different definitions. I was referring to the Peano axioms, there zero is considered a natural number. On the other hand I referred to the fundamental theorem of arithmetic, there it is more convenient to exclude zero so that one can say that every natural number has a unique factorization into primes. And this is perfectly fine, mathematicians can cope with the fact that there are two different sets that different people call the natural numbers.
I think one way to define math is “the collection of symbols we use to precisely explain natural phenomena”.
I can say “that rose is very pink” and that is accurate enough for most people. But if I want to know how much pink there is in the rose, I need a better descriptor than “very”.
So in that sense, Math is made up. It’s just the language we use to describe stuff when spoken/literary languages are imprecise.
But it also isn’t made up because there are objective justified truths behind its symbols. Aliens would probably have the same concepts, they’d just use different symbols.
I can say “that rose is very pink” and that is accurate enough for most people. But if I want to know how much pink there is in the rose, I need a better descriptor than “very”.
So in that sense, Math is made up. It’s just the language we use to describe stuff when spoken/literary languages are imprecise.
But it also isn’t made up because there are objective justified truths behind its symbols. Aliens would probably have the same concepts, they’d just use different symbols.
First of all I would argue that science includes both empirical and analytical sciences, of where math and logic are analytical.
Math and logic is the study of the consequences based on some very simple assumptions and have proven to be extremely valuable to the empirical fields of science.
Second, I would argue that the assumptions are not arbitrary but what most people would hold as true (smartasses and a few more excluded) and even though we have different fields based on what assumptions we choose (fuzzy logic vs classic logic for example) they have all come up with a lot of very usable tools for us for use in the empirical sciences.
Thirdly, "observed" can be argued to both apply to empirical (in the real world) and non-obvious analytical consequences. Sometimes, in analytical science you pull on a thread and see where it leads. The result might surprise you and it is not really wrong to say that you observe the consequence just like you would if you observe a experiment in the empirical sense.
Second, I would argue that the assumptions are not arbitrary but what most people would hold as true (smartasses and a few more excluded) and even though we have different fields based on what assumptions we choose (fuzzy logic vs classic logic for example) they have all come up with a lot of very usable tools for us for use in the empirical sciences.
Thirdly, "observed" can be argued to both apply to empirical (in the real world) and non-obvious analytical consequences. Sometimes, in analytical science you pull on a thread and see where it leads. The result might surprise you and it is not really wrong to say that you observe the consequence just like you would if you observe a experiment in the empirical sense.
As we humans also make up the definitions of all the word, we can certainly be generous and mess a bit with the definition of science to also cover mathematics. Given its importance for the other sciences, it seems a bit mean to exclude it from the club. And if you really want to exclude mathematics, you can simply refer to empirical sciences or whatnot and mathematics will understand. [1]
[1] The Platonists might however disagree.
[1] The Platonists might however disagree.
Are Mathematics and Science both derived from Philosophy? The one uses the other, but seems to be independent.
All of these Philosophy of <discipline> seem to be attempts to bring these exploratory branches of philosophy back to more rigorous roots.
All of these Philosophy of <discipline> seem to be attempts to bring these exploratory branches of philosophy back to more rigorous roots.
Today we generally use the word science to talk about empiricism, i.e. science is composed of theories that haven't been falsified by experiment. But in philosophy, a "science" can be a body of interconnected concepts, propositions, etc that have been established empirically or analytically.
Yes, but without faith in mathematics, you cannot trust scientific results.
It’s an analytic science.