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How has mathematics gotten so abstract?

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Re: How has mathematics gotten so abstract?

#171

Post by hollerith »

fidotron wrote: post #132
hollerith wrote: post #120

Earlier quoted context omitted.

Every mathematician understands what a formal proof is. Ditto a formal statement of a mathematical or logical proposition. The mathematicians of 100 years ago also all understood, and the meaning hasn't changed over the 100 years.

> The mathematicians of 100 years ago also all understood, and the meaning hasn't changed over the 100 years. Isn't that the subject of the whole argument? That mathematicians have taken the road off in a very specific direction, and everyone disagreeing is ejected from the field, rather like occurred more recently in theoretical physics with string theory. Prior to that time quite clearly you had formal proofs which…

>quite clearly you had formal proofs which do not meet the symbolic abstraction requirements

I've been unable to imagine or recall an example. Can you provide one?

Re: How has mathematics gotten so abstract?

#172

Post by zozbot234 »

btilly wrote: post #170
drdeca wrote: post #164

Earlier quoted context omitted.

If such a system proved that the answer to some decidable question was x, when the actual answer was y, then the system would prove a contradiction. If the system doesn’t prove a contradiction, then that situation doesn’t happen, so you can trust its answers to decidable questions. If the only questions you accept as meaningful are the decidable ones, then you can trust its answers for all the questions you accept as…

You're literally talking about how I should regard the hypothetical answers that might be produced by something that I think doesn't exist. There's a pretty clear case of putting the cart before the horse here. On being presumptive about what things can exist, that's the whole point of constructivism. Things only exist when you can construct them. We start with things that everyone accepts, like the natural numbers.…

You can think that something doesn't exist in the general case, while still allowing that it might exist in unspecified narrow cases where additional constraints could apply. For example, there might be algorithms that can decide the halting problem for some non-Turing complete class of programs. Being able to talk in full generality about how such special cases might work is the whole point of non-constructive reasoning. It's "non-constructive" in that it states "I'm not going to construct this just yet".

Re: How has mathematics gotten so abstract?

#173

Post by [deleted] »

goatlover wrote: post #148
abdullahkhalids wrote: post #87

Earlier quoted context omitted.

Mathematical proofs are checked by noisy finite computational machines (humans). Even computer proofs' inputs-outputs are interpreted by humans. Your uncertainty in a theorem is lower bounded by the inherent error rate of human brains.

Plenty of mathematical proofs have been proven true with 100% certainty. Complicated proofs that involve a lot of steps and checking can have errors. They can also be proven true if exhaustively checked.

[deleted]

Re: How has mathematics gotten so abstract?

#174

Post by Ar-Curunir »

falcor84 wrote: post #166
Ar-Curunir wrote: post #125

Earlier quoted context omitted.

No, the correctness of your implementation is a mathematical statement about a computation running a particular computational environment, and can be reasoned about from first principles without ever invoking a computer. Whether your computation gives reasonable outputs on certain inputs says nothing (in general) about the original mathematics.

While mathematics "can" be reasoned about from first principles, the history of math is chock-full of examples of professional mathematicians convinced by unsound and wrong arguments. I prefer the clarity of performing math experiments and validating proofs on a computer.

Yes, but a C or Python program that “implements” a proof and which you test by running it on a few inputs is very different from a program in a interactive theorem prover like Rocq or Lean. In the latter, validity is essentially decided by type-checking, not execution

Re: How has mathematics gotten so abstract?

#175

Post by griffzhowl »

Aardwolf wrote: post #116
pdpi wrote: post #21

Earlier quoted context omitted.

> Pure mathematics is regarded as an abstract science, which it is by definition. I'd argue that, by definition, mathemtatics is not, and cannot be, a science. Mathematics deals with provable truths, science cannot prove truth and must deal falsifiability instead.

I'm not sure if it deals only with provable truths? It even deals with the concept of unprovability itself, if the incompleteness theorem is considered part of mathematics

Yes, but Godel proved the incompleteness theorem, by ingeniously finding ways to prove things about unprovability.

The incompleteness theorem doesn't say that there are statements which are unprovable in any absolute sense. What it says is that given a formal system, there will always be statements which that particular formal system can't prove. But in fact as part of the proof, Godel proves this statement, just not by deriving it in the formal system in question (obviously, since that's what he's proving is impossible).

The way this is done is by using a "metalanguage" to talk about the formal theory in question. In this case it's a kind of ambient set theory. Of course, the proof also implies that if this ambient metalanguage is formalized then there will be sentences which it can't prove either, but these in general will be different sentences for each formalized theory.

Re: How has mathematics gotten so abstract?

#176

Post by Waterluvian »

thadt wrote: post #2

This reminds of of that one time when I was on a date with a girl from the history department who somehow bemusedly sat through my entire mini-lecture on comparing infinite sets. Twenty years and three kids later, she'll still occasionally look me straight in the eye and declare "my infinity is bigger than your infinity."

This is the sweetest thing ever and I hope you feel those butterflies even now sharing this story.

Re: How has mathematics gotten so abstract?

#177

Post by griffzhowl »

Tazerenix wrote: post #13

>Today, mathematics is regarded as an abstract science. Pure mathematics is regarded as an abstract science, which it is by definition . Arnol'd argued vehemently and much more convincingly for the viewpoint that all mathematics is (and must be) linked to the natural sciences. >On forums such as Stack Exchange, trained mathematicians may sneer at newcomers who ask for intuitive explanations of mathematical constructs…

I agree in general but

> Euclid's Elements is 2300 years old and is presented in a completely abstract way.

depends on what you mean by completely abstract. Euclid relies in a logically essential way on the diagrams. Even the first theorem doesn't follow from the postulates as explicitly stated, but relies on the diagram for us to conclude that two circles sharing a radius intersect.

This is a thought-provoking paper on the issue by Viktor Blasjo, Operationalism: An Interpretation of the Philosophy of Ancient Greek Geometry https://link.springer.com/article/10.1007/s10699-021-09791-4

which was recently the subject of a guest video on 3blue1brown https://www.youtube.com/watch?v=M-MgQC6z3VU

Re: How has mathematics gotten so abstract?

#178

Post by drdeca »

tphyahoo2 wrote: post #119

Just drop the axiom of infinity and quit whining. https://en.wikipedia.org/wiki/Ultrafinitism

Can one do QFT in an ultrafinitistic foundations? My guess is no.

Also, I don’t think ZF sans the axiom of infinity works as an ultrafinitistic theory? It still has every natural number, just not the set of all of them.

Re: How has mathematics gotten so abstract?

#179

Post by griffzhowl »

abdullahkhalids wrote: post #87
pdpi wrote: post #21

Earlier quoted context omitted.

> Pure mathematics is regarded as an abstract science, which it is by definition. I'd argue that, by definition, mathemtatics is not, and cannot be, a science. Mathematics deals with provable truths, science cannot prove truth and must deal falsifiability instead.

Mathematical proofs are checked by noisy finite computational machines (humans). Even computer proofs' inputs-outputs are interpreted by humans. Your uncertainty in a theorem is lower bounded by the inherent error rate of human brains.

I agree we can't be absolutely certain of anything (maybe none of our memories are real and we just popped into existence etc.)

But we can be more sure of the deductive validity of a proof than we can be of any of the claims you make in these sentences, so I don't think they can serve to establish any doubt. If we're wrong about deductive logic, then we can only be more wrong about any empirical claims, which rely on deductive logic plus empirical observations

Re: How has mathematics gotten so abstract?

#180

Post by griffzhowl »

nitwit005 wrote: post #130
pdpi wrote: post #21

Earlier quoted context omitted.

> Pure mathematics is regarded as an abstract science, which it is by definition. I'd argue that, by definition, mathemtatics is not, and cannot be, a science. Mathematics deals with provable truths, science cannot prove truth and must deal falsifiability instead.

A proof is just an argument that something is true. Ideally, you've made an extremely strong argument, but it's still a human making a claim something is true. Plenty of published proofs have been shown to be false. Math is scientific in the sense that you've proposed a hypothesis, and others can test it.

The difference is that in mathematics you only have to check the argument. In the empirical sciences you have to both check the argument and also test the conclusion against observations
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