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How has mathematics gotten so abstract?

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Re: How has mathematics gotten so abstract?

#191

Post by btilly »

drdeca wrote: post #188
btilly wrote: post #170

Earlier quoted context omitted.

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.…

> Things only exist when you can construct them. This is exactly what I’m saying is presumptive! If constructivism is to earn the merit of being less presumptive by virtue of not assuming the existence of various things, it should also not assume the non-existence of those things. Which, I think many visions of constructivism do earn this merit, but not your description of it.

So having a different philosophy from you makes me presumptive?

What makes you presume that you have any business telling someone with different beliefs from you, what is OK to believe? You may believe in the existence of whatever you like. Whether that be numbers that cannot be specified, or invisible pink unicorns.

I'll be over in the corner saying that your belief does not compel me to agree with you on the question of what exists. Not when your belief follows from formalism, which explicitly abandons any pretense of meaningfulness to its abstract symbol manipulation.

Re: How has mathematics gotten so abstract?

#192

Post by zozbot234 »

drdeca wrote: post #188
btilly wrote: post #170

Earlier quoted context omitted.

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.…

> Things only exist when you can construct them. This is exactly what I’m saying is presumptive! If constructivism is to earn the merit of being less presumptive by virtue of not assuming the existence of various things, it should also not assume the non-existence of those things. Which, I think many visions of constructivism do earn this merit, but not your description of it.

The underlying problem is that constructivism and non-constructive reasoning are using the word "exists" (and, relatedly, the logical disjunction) to mean very different things. The constructive meaning for "exists" is certainly more intuitive, so it makes sense that constructivists would want it by 'default'; but the non-constructive operator (which a constructivist would preferably understand as "is merely allowed to exist"), while somewhat more subtle, has a usefulness of its own.

Re: How has mathematics gotten so abstract?

#193

Post by Jensson »

griffzhowl wrote: post #180
nitwit005 wrote: post #130

Earlier quoted context omitted.

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

> In the empirical sciences you have to both check the argument and also test the conclusion against observations

That isn't true, you just test new axioms but most stuff we do in empirical sciences don't require new axioms.

The only difference between material sciences and math is that in math you don't test axioms while in empirical sciences you do.

Re: How has mathematics gotten so abstract?

#194

Post by Jensson »

anthk wrote: post #165
stonemetal12 wrote: post #60

Earlier quoted context omitted.

Mathematics arose from ancient humans need to count and measure. Even the invention\discovery of Calculus was in service to physics. It has probably only been 300 years or so since Mathematics has been symbolic, before that it was more geometric and more attached to the physical world. Leibniz (late 1600s) helped to popularize negative numbers. At the time most mathematicians thought they were "absurd" and "fictitiou…

Archimedes did Calculus before Newton. https://en.wikipedia.org/wiki/The_Method_of_Mechanical_Theor...

He didn't connect the dots, so no he didn't do calculus even if he did some things related to it.

Re: How has mathematics gotten so abstract?

#195

Post by edanm »

btilly wrote: post #145
edanm wrote: post #114

Earlier quoted context omitted.

> When you say "given ZFC", you're assuming a lot. Errr, I'm just assuming the axioms of ZFC. That's literally all I'm doing. > In what sense do [numbers that can't be finitely specified] exist? In the sense that we can describe rules that lead to them, and describe how to work with them. I understand that you're trying to tie the notion of "existence" to constructability, and that's fine. That's one way to play the…

My point is that going from a lay understanding of mathematics to "just accept ZFC" means jumping past a variety of debatable philosophical points, and accepting a standard collection of answers to them. Mathematicians gloss over that.

Yeah, I think that's fair.

On the other hand, I think it's really cool to teach laypeople about things like "sizes of infinities", etc. They are deep math concepts that can be taught with relatively simple analogies that most people understand, and they're interesting things to know. I know that I personally loved learning about them as a kid, before I had almost any knowledge of math - it's one of the reasons that while I initially didn't connect with other areas of math, I found set theory delightful as a kid.

I just feel like if you need to first walk people through a bunch of philosophical back and forth on constructionism, you'll never get to the fun stuff.

Re: How has mathematics gotten so abstract?

#196

Post by edanm »

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.…

> Therefore an oracle to decide the Halting problem does not exist within the constructible universe.

I might be confused here, but isn't an Oracle to decide the halting problem something that everyone agrees doesn't exist?

The whole idea is for this to be a thought experiment. "If we magically had a way to decide the halting problem, how would that affect things" seems like a normal hypothetical question.

Re: How has mathematics gotten so abstract?

#197

Post by zozbot234 »

edanm wrote: post #196
btilly wrote: post #170

Earlier quoted context omitted.

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.…

> Therefore an oracle to decide the Halting problem does not exist within the constructible universe. I might be confused here, but isn't an Oracle to decide the halting problem something that everyone agrees doesn't exist? The whole idea is for this to be a thought experiment. "If we magically had a way to decide the halting problem, how would that affect things" seems like a normal hypothetical question.

Everyone agrees that you can't write an algorithm for a Turing machine (or computational equivalent) that decides the Halting problem for Turing machines in every case. Since this is explicitly worded as "you can't write an algorithm for..." it's in fact talking about a kind of constructive existence and saying that it doesn't apply. The oracle concept is normally phrased about "If we magically had a way to decide this undecidable problem in every case" but it's real utility from a constructive POV is talking about special cases that you haven't bothered to narrow down just yet.

Re: How has mathematics gotten so abstract?

#198

Post by tempfile »

ogogmad wrote: post #127
tempfile wrote: post #111

Earlier quoted context omitted.

You are being very cryptic. Are you trying to say that the existence of uncountable sets requires the axiom of choice? If you are, that's false. If you aren't, I'm not sure what you are trying to say.

He never mentioned the Axiom of Choice. I think he articulated his opinion clearly enough. It's his own subjective value judgement.

I don't think either of us think what he wrote is subjective or an opinion. they seem like pretty definitely truth claims to me.

Re: How has mathematics gotten so abstract?

#199

Post by btilly »

edanm wrote: post #195
btilly wrote: post #145

Earlier quoted context omitted.

My point is that going from a lay understanding of mathematics to "just accept ZFC" means jumping past a variety of debatable philosophical points, and accepting a standard collection of answers to them. Mathematicians gloss over that.

Yeah, I think that's fair. On the other hand, I think it's really cool to teach laypeople about things like "sizes of infinities", etc. They are deep math concepts that can be taught with relatively simple analogies that most people understand, and they're interesting things to know. I know that I personally loved learning about them as a kid, before I had almost any knowledge of math - it's one of the reasons that w…

We each find different things delightful. What I like, you may not. And vice versa.

But it is easy to present deep ideas from constructivism, without mentioning the word constructivism. Or even acknowledging that the philosophy exists.

For example the second half of https://math.stackexchange.com/questions/5074503/can-pa-prov... is an important constructivist thing. It shows why everything that a constructivist could ever be interested in mathematically, can be embedded in the natural numbers. With all of the constructions needing nothing more than the Peano Axioms. (Proving the results may need stronger axioms though...)

From my point of view, https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach does something similar. That book got a lot of people interested in basic concepts around recursion, computation, and what it means to think. Absolutely everything in it works constructively. And yet that philosophy is not mentioned. Not even once.

The only point where a constructivist need discuss all of the philosophical back and forth on constructivism, is in explaining why a constructivist need not accept various claims coming out of classical mathematics. And even that discussion would not be so painful if people who have learned classical mathematics were more aware of the philosophical assumptions that they are making.

Re: How has mathematics gotten so abstract?

#200

Post by btilly »

edanm wrote: post #196
btilly wrote: post #170

Earlier quoted context omitted.

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.…

> Therefore an oracle to decide the Halting problem does not exist within the constructible universe. I might be confused here, but isn't an Oracle to decide the halting problem something that everyone agrees doesn't exist? The whole idea is for this to be a thought experiment. "If we magically had a way to decide the halting problem, how would that affect things" seems like a normal hypothetical question.

You literally cannot doubt the existence of this oracle, without doubting what existence means in classical mathematics.

Here is why a classical mathematician would say that this oracle exists.

Let f(program, input, n) be 1 or 0 depending on whether the program program, given input input, is still running at step n. This is a perfectly well-behaved mathematical function. In fact it is a computable one - we can compute it by merely running a simulation of a computer for a fixed number of steps.

Let oracle(program, input) be the limit, as n goes to infinity, of f(program, input, n). Classically this limit always exists, and always gives us 0 or 1. The fact that we happen to be unable to compute it, doesn't change the fact that this is a perfectly well-defined function according to classical mathematics.

If you give up the existence of this oracle, you might as well give up the existence of any real numbers that do not have a finite description. Which is to say, almost all of them. Why? Because the set of finite descriptions is countable, and therefore the set of real numbers that admit a finite description is also only countable. But there are an uncountable number of real numbers, so almost all real numbers do not admit a finite description.

The real question isn't whether this oracle exists. It is what you want the word "exists" to mean.

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