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How has mathematics gotten so abstract?

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Re: How has mathematics gotten so abstract?

#101

Post by edanm »

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.

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?

Re: How has mathematics gotten so abstract?

#102

Post by Etherlord87 »

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.

You don't need an implicit philosophical assumption, you just need to define what an infinity is and the comparison method.

Re: How has mathematics gotten so abstract?

#103

Post by btilly »

dragonwriter wrote: post #91
btilly wrote: post #89

Earlier quoted context omitted.

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.

> 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. That’s not at all true at the level where you are dealing with different infinities, usually, which tends to come after the (usually, fairly early) part dealing with proofs and the fact that all mathematics is dealing with “unprovable intellec…

I guarantee that a naive presentation doesn't actually include the axioms, and doesn't address the philosophical questions dividing formalism from constructivism.

Uncountable need not mean more. It can mean that there are things that you can't figure out whether to count, because they are undecidable.

Re: How has mathematics gotten so abstract?

#104

Post by btilly »

edanm wrote: post #101
btilly wrote: post #89

Earlier quoted context omitted.

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.

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 weirder things. Such as well-specified finite problems that provably have a polynomial time algorithm to solve...but for which it is impossible to find or verify that algorithm, or put an upper bound on the constants in the algorithm. In what sense does that algorithm exist, and is finite?

Does that sound impossible? An example of an open problem whose algorithm may have those characteristics is an algorithm to decide which graphs can be drawn on a torus without any self-crossings.

If our notion of "exists" is "constructable", all possible mathematical things can fit inside of a countable universe. No set can have more than that.

Re: How has mathematics gotten so abstract?

#105

Post by 2snakes »

Tazerenix wrote: post #53
TimPC wrote: post #37

Earlier quoted context omitted.

In general you aren't testing as an empiricist though, you are looking for a rational argument to prove or disprove something.

The practical experience of doing mathematics is actually quite close to a natural science, even if the subject is technically a "formal science* according to the conventional meanings of the terms. Mathematicians actually do the same thing as scientists: hypothesis building by extensive investigation of examples. Looking for examples which catch the boundary of established knowledge and try to break existing assumpt…

An alternative to abstraction is to use iconic forms and boundary math (containerization and void-based reasoning). See Laws of Form and William Bricken's books recently. Using a unary operator instead of binary (Boolean) does indeed seem simpler, in keeping with Nature. Introduction: https://www.frontiersin.org/journals/psychology/articles/10....

Re: How has mathematics gotten so abstract?

#106

Post by btilly »

Etherlord87 wrote: post #102
btilly wrote: post #89

Earlier quoted context omitted.

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.

You don't need an implicit philosophical assumption, you just need to define what an infinity is and the comparison method.

Here's a hint. When someone makes a reference to something that was actively debated for decades, and you're not familiar with said debates, you should probably assume that you're missing some piece of relevant knowledge.

https://plato.stanford.edu/entries/mathematics-constructive/ is one place that you could start filling in that gap.

Re: How has mathematics gotten so abstract?

#107

Post by Warwolt »

weinzierl wrote: post #62
lo_zamoyski wrote: post #57

Earlier quoted context omitted.

It is absolutely a science, a formal science. What it isn't is an empirical science. The "symbol pushing" is a methodological tool, and a very useful one that opened up the possibility of new expansive fields of mathematics. (Of course, it is important to always distinguish between properties of the abstraction or the tool from the object of study.)

Well, we are talking about pure mathematics and there is not much Popperian scientific method in it.

Who cares? That's just semantics. If we define science as the systematic search for truths, then mathematics and logic are the paradigmic sciences. If we define it as only empirical search for truth then perhaps that excludes mathematics, but it's an entirely unintersting point, since it says nothing.

Re: How has mathematics gotten so abstract?

#108

Post by thadt »

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.

Probably not. But this one time we had an argument and I made a statement along the lines of "I'm right, naturally." She went irrational. I lost the argument.

QED

Re: How has mathematics gotten so abstract?

#109

Post by btilly »

thadt wrote: post #108
btilly wrote: post #89

Earlier quoted context omitted.

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.

Probably not. But this one time we had an argument and I made a statement along the lines of "I'm right, naturally." She went irrational. I lost the argument. QED

LOL

If she laughs at that kind of thing, I can see why you married her.

Re: How has mathematics gotten so abstract?

#110

Post by azan_ »

fidotron wrote: post #48
azan_ wrote: post #31

Earlier quoted context omitted.

Theirs no such thing as excessive abstraction in math, because abstraction is the point. Is category theory “excessive abstraction” in your opinion?

> 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.
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