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How has mathematics gotten so abstract?

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Re: How has mathematics gotten so abstract?

#111

Post by tempfile »

btilly wrote: post #104
edanm wrote: post #101

Earlier quoted context omitted.

What leaps are "unprovable"? I'm curious, that doesn't sound right. For sure there are valid arguments on whether or not to use certain axioms which allow or disallow some set theoretical constructions, but given ZFC, is there anything that follows that is unprovable?

When you say "given ZFC", you're assuming a lot. Including a notion of mathematical existence which bears little relation to any concept that most lay people have of what mathematical existence might mean. In particular, you have made sufficient assumptions to prove that almost all real numbers that exist can never be specified in any possible finite description. In what sense do they exist? You also wind up with wei…

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.

Re: How has mathematics gotten so abstract?

#112

Post by somecontext »

btilly wrote: post #104
edanm wrote: post #101

Earlier quoted context omitted.

What leaps are "unprovable"? I'm curious, that doesn't sound right. For sure there are valid arguments on whether or not to use certain axioms which allow or disallow some set theoretical constructions, but given ZFC, is there anything that follows that is unprovable?

When you say "given ZFC", you're assuming a lot. Including a notion of mathematical existence which bears little relation to any concept that most lay people have of what mathematical existence might mean. In particular, you have made sufficient assumptions to prove that almost all real numbers that exist can never be specified in any possible finite description. In what sense do they exist? You also wind up with wei…

> you're assuming a lot. Including a notion of mathematical existence which bears little relation to any concept that most lay people have of what mathematical existence might mean.

John Horton Conway:

> It's a funny thing that happens with mathematicians. What's the ontology of mathematical things? How do they exist? In what sense do they exist? There's no doubt that they do exist but you can't poke and prod them except by thinking about them. It's quite astonishing and I still don't understand it, having been a mathematician all my life. How can things be there without actually being there? There's no doubt that 2 is there or 3 or the square root of omega. They're very real things. I still don't know the sense in which mathematical objects exist, but they do. Of course, it's hard to say in what sense a cat is out there, too, but we know it is, very definitely. Cats have a stubborn reality but maybe numbers are stubborner still. You can't push a cat in a direction it doesn't want to go. You can't do it with a number either.

Re: How has mathematics gotten so abstract?

#113

Post by fidotron »

azan_ wrote: post #110
fidotron wrote: post #48

Earlier quoted context omitted.

> because abstraction is the point. Formal reasoning is the point, which is not by itself abstraction. Someone else in this discussion is saying Euclid's Elements is abstract, which is near complete nonsense. If that is abstract our perception of everything except for the fundamental [whatever] we are formed of is an abstraction.

No, abstraction is the point and formal reasoning is a tool. And yes, what Euclid did is obviously abstraction, I don’t know why so you consider this stance nonsense.

Can you say how mathematics is inherently abstract in a way consistent with your day-to-day life as a concrete person? Or is your personhood also an abstraction?

I could construct a formal reasoning scheme involving rules and jugs on my table, where we can pour liquids from one to another. It would be in no way symbolic, since it could use the liquids directly to simply be what they are. Is constructing and studing such a mechanism not mathematics? Similarly with something like musical intervals.

Re: How has mathematics gotten so abstract?

#114

Post by edanm »

btilly wrote: post #104
edanm wrote: post #101

Earlier quoted context omitted.

What leaps are "unprovable"? I'm curious, that doesn't sound right. For sure there are valid arguments on whether or not to use certain axioms which allow or disallow some set theoretical constructions, but given ZFC, is there anything that follows that is unprovable?

When you say "given ZFC", you're assuming a lot. Including a notion of mathematical existence which bears little relation to any concept that most lay people have of what mathematical existence might mean. In particular, you have made sufficient assumptions to prove that almost all real numbers that exist can never be specified in any possible finite description. In what sense do they exist? You also wind up with wei…

> 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 game. Another is to use ZFC and be fine with "weird, unintuitive to laypeople" outcomes. Both are interesting and valid things to do IMO. I'm just not sure why one is obviously "better" or "more real" or something. At the end, it's all just coming up with rules and figuring out what comes out of them.

Re: How has mathematics gotten so abstract?

#115

Post by azan_ »

fidotron wrote: post #113
azan_ wrote: post #110

Earlier quoted context omitted.

No, abstraction is the point and formal reasoning is a tool. And yes, what Euclid did is obviously abstraction, I don’t know why so you consider this stance nonsense.

Can you say how mathematics is inherently abstract in a way consistent with your day-to-day life as a concrete person? Or is your personhood also an abstraction? I could construct a formal reasoning scheme involving rules and jugs on my table, where we can pour liquids from one to another. It would be in no way symbolic, since it could use the liquids directly to simply be what they are. Is constructing and studing s…

Of course I can. I frequently use numbers which are great abstraction. I can use same number five to describe apples, bananas and everything countable.

Re: How has mathematics gotten so abstract?

#116

Post by Aardwolf »

pdpi wrote: post #21
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…

> 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

Re: How has mathematics gotten so abstract?

#117

Post by fidotron »

azan_ wrote: post #115
fidotron wrote: post #113

Earlier quoted context omitted.

Can you say how mathematics is inherently abstract in a way consistent with your day-to-day life as a concrete person? Or is your personhood also an abstraction? I could construct a formal reasoning scheme involving rules and jugs on my table, where we can pour liquids from one to another. It would be in no way symbolic, since it could use the liquids directly to simply be what they are. Is constructing and studing s…

Of course I can. I frequently use numbers which are great abstraction. I can use same number five to describe apples, bananas and everything countable.

> to describe apples, bananas and everything countable

An apple is an abstraction over the particles/waves that comprise it, as is a banana.

Euclid is no more abstract than the day to day existence of a normal person, hence to claim that it is unusually abstract is to ignore, as you did, the abstraction inherent in day to day life.

As I pointed out it's very possible to create formal reasoning systems which are not symbolic or abstract, but due to that are we to assume constructing or studying them would not be a mathematical exercise? In fact the Pythagoreans did all sorts of stuff like that.

Re: How has mathematics gotten so abstract?

#118

Post by N0isRESFe8GXmqR »

btilly wrote: post #89
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."

I'm curious. Did either of you ever notice the implicit philosophical assumptions that you have to make to come to the conclusion that one infinity can be larger than another? Despite the fact that this was actively debated for decades, modern math courses seldom acknowledge the fact that they are making unprovable intellectual leaps along the way.

[dead]

Re: How has mathematics gotten so abstract?

#120

Post by hollerith »

fidotron wrote: post #86
empath75 wrote: post #83

Earlier quoted context omitted.

I am not, this is what formal logic and formal reasoning means: https://plato.stanford.edu/entries/logic-classical/ "Formal" in logic has a very precise technical meaning.

What you mean is someone has redefined the word to suit their purpose, which is precisely what I pointed out at the top. Edit to add: this comment had a sibling, that was suggesting that given a specific proof assistant requires all input to be formal logic perhaps the word formal could be redefined to mean that which is accepted by the proof assistant. Sadly this fine example of my point has been deleted.

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.
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