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How has mathematics gotten so abstract?

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Re: How has mathematics gotten so abstract?

#131

Post by gnulinux »

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."

Wow, I did a very similar thing on the first date with my now wife. I explained the halting problem, and Godel's incompleteness theorems. We also talked about her (biomedical) research, so it wasn't a one sided conversation.

I think dominating on a first date is a risk (which I was mindful of) but just being yourself, and talking about something you're truly passionate about is the key.

Re: How has mathematics gotten so abstract?

#132

Post by fidotron »

hollerith wrote: post #120
fidotron wrote: post #86

Earlier quoted context omitted.

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.

> 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 do not meet the symbolic abstraction requirements that pure mathematicians apparently believe are axiomatic to their field today, even if they attempt to pretend otherwise, as argued over the case of Euclid elsewhere. If the Pythagoreans were reincarnated, as they probably expected, they would no doubt be dismissed as crackpots by these same people.

Re: How has mathematics gotten so abstract?

#133

Post by zozbot234 »

btilly wrote: post #103
dragonwriter wrote: post #91

Earlier quoted context omitted.

> 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.

The "philosophical questions" dividing formalism from constructivism are greatly overstated. The point of having those degrees of undecidability or uncountability is precisely to be able to say things like "even if you happen to be operating under strong additional assumptions that let you decide/count X, that still doesn't let you decide/count Y in general." That's what formalism is: a handy way of making statements about what you can't do constructively in the general case.

To be fair, constructivists tend to prefer talk about different "universes" as opposed to different "sizes" of sets, but that's all it is: little more than a mere difference in terminology! You can show equiconsistency statements across these different points of view.

Re: How has mathematics gotten so abstract?

#134

Post by susam »

This article explores a particular kind of abstractness in mathematics, especially the construction of numbers and the cardinalities of infinite sets. It is all very interesting indeed.

However, the kind of abstractness I most enjoy in mathematics is found in algebraic structures such as groups and rings, or even simpler structures like magmas and monoids. These structures avoid relying on specific types of numbers or elements, and instead focus on the relationships and operations themselves. For me, this reveals an even deeper beauty, i.e., different domains of mathematics, or even problems in computer science, can be unified under the same algebraic framework.

Consider, for example, the fact that the set of real numbers forms a vector space over the set of rationals. Can it get more abstract than that? We know such a vector space must have a basis, but what would that basis even look like? The existence of such a basis (Hamel basis) is guaranteed by the axioms and proofs, yet it defies explicit description. That, to me, is the most intriguing kind of abstractness!

Despite being so abstract, the same algebraic structures find concrete applications in computing, for example, in the form of coding theory. Concepts such as polynomial rings and cosets of subspaces over finite fields play an important role in error-correcting codes, without which modern data transmission and storage would not exist in their current form.

Re: How has mathematics gotten so abstract?

#135

Post by zozbot234 »

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…

> In what sense do they exist?

In the sense that all statements of non-constructive "existence" are made, viz. "you can't prove that they don't exist in the general case", so you are allowed to work under the stronger assumption that they also exist constructively, without any contradiction resulting. That can certainly be useful in some applications.

Re: How has mathematics gotten so abstract?

#136

Post by trinsic2 »

>Next, consider the time needed for Achilles to reach the yellow dot; once again, by the time he gets there, the turtle will have moved forward a tiny bit. This process can be continued indefinitely; the gap keeps getting smaller but never goes to zero, so we must conclude that Achilles can’t possibly win the race.

Am i daft, eventually (Very soon) Achilles would over take the turtles position regardless of how far it moved... I am missing something?

Re: How has mathematics gotten so abstract?

#137

Post by m_dupont »

trinsic2 wrote: post #136

>Next, consider the time needed for Achilles to reach the yellow dot; once again, by the time he gets there, the turtle will have moved forward a tiny bit. This process can be continued indefinitely; the gap keeps getting smaller but never goes to zero, so we must conclude that Achilles can’t possibly win the race. Am i daft, eventually (Very soon) Achilles would over take the turtles position regardless of how far i…

you're not, the proof is a famous error known as zenos paradox. Its only an apparent paradox, and indeed it's been disproven by observing that things do in fact move

Re: How has mathematics gotten so abstract?

#139

Post by prmph »

jjgreen wrote: post #25

The number 1 is what a cow, a fox, a stone ... have in common, oneness. Mathematics is abstraction, written down.

That's not obvious.

- they are material objects

- they are concepts I understand

- they are sequences of letters

- they are English words

- ...

Not sure why oneness is privileged as what they have in common, and their oneness is meaningless by itself. Oneness is a property that is only meaningful in relation to other concepts of objects.

Re: How has mathematics gotten so abstract?

#140

Post by btilly »

godelski wrote: post #128
btilly wrote: post #103

Earlier quoted context omitted.

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.

> I guarantee that a naive presentation doesn't actually include the axioms But you said "modern math courses". Are you now talking about a casual conversation? I mean the OP's story is that his wife just liked listening to him talk about his passions. > Uncountable need not mean more. Sure. But that doesn't mean that there aren't differing categories. However you slice it, we can operate on these things in different…

Your guesses at what I seem to think are completely off base and insulting.

When I say "modern math courses", I mean like the standard courses that most future mathematicians take on their way to various degrees. For all that we mumble ZFC, it is darned easy to get a PhD in mathematics without actually learning the axioms of ZFC. And without learning anything about the historical debates in the foundations of mathematics.

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