Philosophy of Mathematics and Metamathematics (1 Viewer)

[Sy123]

The philosophy of mathematics asks questions about the foundations of mathematics, validity of certain truths in mathematics, it's assumptions, implications, the correspondence of mathematics and reality and so on.

One primary question to ask is whether mathematics is a subject that refers to real mathematical entities, these entities are real and objective, and the job of the mathematician is to discover these entities and their relationships to each other. This is what is known as "mathematical realism", in opposition is "mathematical anti-realism", which seeks to ground mathematics not in 'reality' but in something different.

One way in which anti-realism was promoted was with what is called 'formalism', it is the idea that mathematics is simply a game in language, of manipulating symbols according to various symbolic rules (axioms), and then getting symbolic conclusions. These symbols are merely that, meaningless (meaningless insofar we are talking about objective reality) constructions of mathematicians. The most popular realist position on mathematics is mathematical platonism, which proposes that mathematical objects are really existing entities just like physical objects, like physical objects, mathematical objects are not reducible to mental entities and the imagination.

Where do you stand on this?

I wonder if we have any constructivists here who reject existence proofs that do not involve construction (if so, my (curious) question is how they prove the existence of the irrational numbers, as far as I can see, normal proofs for irrationality of certain numbers are not constructive, i.e. proof that sqrt(2) is irrational)

 

[leehuan]

I just woke up from a nap so this was a bit fuzzy, and so would be what follows. But I'll just answer with a debate topic I was asked quite a few months back: That maths was discovered, not invented.

The behaviour of the world around us may appear random and spontaneous, however that becomes a further topic. I believe our 'language' of mathematics is essentially our way to convey how we perceive the world around us (which means, it would technically be a discovery). One may argue that our perceptions can be completely flawed, however we have the power to redefine any elements that we find false. That can be said for physics as well.

If you wish to consider maths as visible and tangible, then I would say no maths is not physical. If you want to tell me maths is a medium which we explain properties that are visible and tangible, I would say it is.

 

[glittergal96]

I firmly stand on the formalist side of the fence mostly. Although our choice of axioms for our formal logical systems are motivated by patterns witnessed in science/nature/human experience.

In that sense many of the formal mathematical objects (eg natural numbers) make a lot of intuitive sense to humans and reflect some sort of apparent structure of the universe. But even when we talk about things like real numbers we are delving into abstractions that imo do not exist in any physical sense. Any measurements we ever take will be rational etc. The insufficiencies of the rational numbers are only really clear in the abstract perfect world of mathematics where we can construct perfect isosceles right angled triangles, and measure an abstract property called "length" of a circle's diameter vs circumference.

Also, the proofs of a form like this: https://en.wikipedia.org/wiki/Square_root_of_2#Constructive_proof do not use the law of the excluded middle, and hence are valid in most constructivist systems. Note that definitions of objects are necessarily different in a constructivist world. Eg, being irrational is a stronger property than being a real number that is not a rational number. It means being a real number a such that you can prove |a-x|>p for each rational number x, where p is a positive real number depending on x.

So not being rational is NOT enough to imply that you satisfy that inequality in constructivist systems, which might seem bizarre to most of you.

Thankfully, the wiki link demonstrates that we can prove that the square root of 2 is not only a real number that is not rational, moreover it is an irrational number!

The standard proof by contradiction still suffices to prove the weaker fact that the square root of 2 is not rational.

Note that arguments of the form

1. p leads to a contradiction -> ~p

are valid in intuitionist logic, but

2. ~p leads to a contradiction -> p

are not. (We need the law of the excluded middle to prove ~~p->p and/or p->~~p, from which the equivalence of the above assertions would follow.)

 

[Sy123]

I firmly stand on the formalist side of the fence mostly. Although our choice of axioms for our formal logical systems are motivated by patterns witnessed in science/nature/human experience.

In that sense many of the formal mathematical objects (eg natural numbers) make a lot of intuitive sense to humans and reflect some sort of apparent structure of the universe. But even when we talk about things like real numbers we are delving into abstractions that imo do not exist in any physical sense. Any measurements we ever take will be rational etc. The insufficiencies of the rational numbers are only really clear in the abstract perfect world of mathematics where we can construct perfect isosceles right angled triangles, and measure an abstract property called "length" of a circle's diameter vs circumference.

Need measurements that are infinitely precise be possible if mathematical entities exist? It seems to me that they are different questions.

Also, the proofs of a form like this: https://en.wikipedia.org/wiki/Square_root_of_2#Constructive_proof do not use the law of the excluded middle, and hence are valid in most constructivist systems. Note that definitions of objects are necessarily different in a constructivist world. Eg, being irrational is a stronger property than being a real number that is not a rational number. It means being a real number a such that you can prove |a-x|>p for each rational number x, where p is a positive real number depending on x.

So not being rational is NOT enough to imply that you satisfy that inequality in constructivist systems, which might seem bizarre to most of you.

Thankfully, the wiki link demonstrates that we can prove that the square root of 2 is not only a real number that is not rational, moreover it is an irrational number!

The standard proof by contradiction still suffices to prove the weaker fact that the square root of 2 is not rational.

Note that arguments of the form

1. p leads to a contradiction -> ~p

are valid in intuitionist logic, but

2. ~p leads to a contradiction -> p

are not. (We need the law of the excluded middle to prove ~~p->p and/or p->~~p, from which the equivalence of the above assertions would follow.)

Right I see that argument, I had arguments of the form that seek to prove that if p/q = sqrt(2) then p and q are both even which yields contradiction, in mind when thinking of non-constructive proofs, but that seems to be a nice constructive one.

 

[Sy123]

I'd notably mention that in UNSW itself there is the "Sydney School" of the philosophy of mathematics, namely an Aristotelian realist approach to Mathematics

http://web.maths.unsw.edu.au/~jim/structmath.html

 

[glittergal96]

Need measurements that are infinitely precise be possible if mathematical entities exist? It seems to me that they are different questions.

Right I see that argument, I had arguments of the form that seek to prove that if p/q = sqrt(2) then p and q are both even which yields contradiction, in mind when thinking of non-constructive proofs, but that seems to be a nice constructive one.

They are indeed, and I was not providing a formal argument there but more a vague justification for my personal point of view (I do not think a convincing argument could be made either way on this matter). Many people might think of real numbers as more tangible and observable in our world than they are by mistaking mathematical models of physical objects (a unit circle) with the physical objects themselves (eg a wheel made out of a finite number of atoms of metallic elements, which is really rough on the edges if you zoom in sufficiently). There is no physically manifest example (that I can think of) of a perfect circle. Even if we were to think of the integers as "existing" in some sense because they seem to perfectly capture the human notion of cardinality, we still wouldn't be able to use this kind of (dodgy) reasoning to talk about the reals at all.

I know, and this is the standard proof I mentioned in my previous post. This is still valid within intuitionist logic but it only proves the weaker assertion I mentioned in my previous post. Proofs "by contradiction" are still often okay in constructivist mathematics. It tends to be things like proofs of existence of things by contradiction that are illegal.

Tl;dr They might exist, they might not. But however mathematicians intuit upon their work and whatever deeper truth there may be in their theorems, the rules of the game that they actually play are those of formal logic.