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Numbers Are Leaves

christo.sh

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Re: Numbers Are Leaves

#61

Post by ludston »

md224 wrote: post #52

> set theory is the foundation of all of mathematics I disagree. I would say set theory is a foundation, not the foundation. Which system is the "correct" foundation of mathematics? Does it even make sense to talk about correctness in this context? These are open questions and they're very interesting! Don't prematurely close yourself off to them by assuming that set theory's role is some kind of scientific fact.

Kurt Godel kind of threw this line of reasoning into the bin unfortunately. No system can be both complete and consistent, therefore the authors statement that the set theory he is studying is the basis of all mathematics as well as consistent is probably false.

Re: Numbers Are Leaves

#62

Post by mondrian »

photonthug wrote: post #32

A related rabbit hole that you can jump down in FoM is that Zermelo's ordinals and von Neumann ordinals cannot both be true at the same time. Wikipedia's intro to the topic [0] is a starting place, see also [1] which might be more in depth. [0] https://en.m.wikipedia.org/wiki/Benacerraf%27s_identificatio... [1] https://plato.stanford.edu/entries/philosophy-mathematics/#W... > If you don't know why set theory is impor…

That’s pretty interesting, thanks. It’s related to abstraction (interfaces) vs representation (implementation) in programming. To my eye, there’s no conflict that 1 \in 3 is true in one representation and not in the other. Trying to use \in like that seems like a violation of an abstract interface, somehow expecting that the various implementations of an abstraction must be identical. It also doesn’t seem to violate platonism, since “numbers” are abstract ideas that are not expressible directly in set theory. Set theory can only encode concrete representations of numbers. Much like any physical chair cannot be identical to the platonic Chair.

Re: Numbers Are Leaves

#63

Post by DiscourseFan »

kevinventullo wrote: post #59
TZubiri wrote: post #21

Earlier quoted context omitted.

If 1 out of 10 chmessicians discover something useful, the discoverer had good taste and deserves credit. There is no insurance redistributing credit amongst all of the pointless searches. The guys who invented imaginary numbers or eigenvectors weren't just throwing darts at a board and got "lucky".

It sounds like you have an infallible instinct for exactly which lines of research should be funded and which are useless dead ends. The NSF should hire you immediately!

always the problem with scientific research is that we never know why anything actually works the way it does, but we have a lot of ways of talking about it

Re: Numbers Are Leaves

#64

Post by [deleted] »

kittikitti wrote: post #55
isoprophlex wrote: post #10

This has almost nothing to do with numbers or ZFC and almost everything to do with how graph layout algo's produce their outputs...

I agree. The layout makes it difficult to point out the node in the set that actually represents the graph. In the layout proposed, I'm not sure which node is 5 from the graph representing 5, unless someone can clue me in?

[deleted]

Re: Numbers Are Leaves

#65

Post by mymoomin »

ludston wrote: post #61
md224 wrote: post #52

> set theory is the foundation of all of mathematics I disagree. I would say set theory is a foundation, not the foundation. Which system is the "correct" foundation of mathematics? Does it even make sense to talk about correctness in this context? These are open questions and they're very interesting! Don't prematurely close yourself off to them by assuming that set theory's role is some kind of scientific fact.

Kurt Godel kind of threw this line of reasoning into the bin unfortunately. No system can be both complete and consistent, therefore the authors statement that the set theory he is studying is the basis of all mathematics as well as consistent is probably false.

The article is right to say that set theory can serve as a foundation for almost all other mathematics, and you're also right to say that no reasonably-complex consistent system of axioms can be complete. The resolution to this is that if you ground something (let's say topology) in e.g. ZFC (the most commonly used system of axioms for set theory) then incompleteness in ZFC maps to incompleteness in topology. Here's an example https://en.wikipedia.org/wiki/Moore_space_(topology)#Normal_... .

There are other foundations, some of which are based on things other than set theory (category theory, type theory), but they're usually equivalent to ZFC ± a few axioms, because you can embed those other foundations in some kind of set theory, and embed set theory in the other foundations.

Re: Numbers Are Leaves

#67

Post by monadINtop »

kevinventullo wrote: post #59
TZubiri wrote: post #21

Earlier quoted context omitted.

If 1 out of 10 chmessicians discover something useful, the discoverer had good taste and deserves credit. There is no insurance redistributing credit amongst all of the pointless searches. The guys who invented imaginary numbers or eigenvectors weren't just throwing darts at a board and got "lucky".

It sounds like you have an infallible instinct for exactly which lines of research should be funded and which are useless dead ends. The NSF should hire you immediately!

But he's not wrong, research really isn't just random playing (which I think is obviously still good and fun and should be done for it's own sake, of which this blog post is a great example though it probably wouldn't be worth the time of a formal research project).

It's the reason why "good questions" and seemingly arbitrary or trivial problems - especially those which motivates the development of much deeper machinery or discovery in order to solve them - is widely appreciated across all fields of math (poincare conjecture, galoi's proof of the unsolvability of the quintic, fermat's last theorem, riemann zeta's zeroes etc.)

In all cases those good questions where not randomly cooked up but posed from a previous, more direct line of inquiry, of which the originator usually had a good insight into. Though for the examples I gave the unexpected depth certainly could not have been anticipated beforehand.

Re: Numbers Are Leaves

#69

Post by Viliam1234 »

kevinventullo wrote: post #59
TZubiri wrote: post #21

Earlier quoted context omitted.

If 1 out of 10 chmessicians discover something useful, the discoverer had good taste and deserves credit. There is no insurance redistributing credit amongst all of the pointless searches. The guys who invented imaginary numbers or eigenvectors weren't just throwing darts at a board and got "lucky".

It sounds like you have an infallible instinct for exactly which lines of research should be funded and which are useless dead ends. The NSF should hire you immediately!

People can make a good educated guess without being literally infallible.

Re: Numbers Are Leaves

#70

Post by Viliam1234 »

bubblyworld wrote: post #37
nthingtohide wrote: post #16

I think people will like the following tangent. https://en.m.wikipedia.org/wiki/Benacerraf%27s_identificatio... In the philosophy of mathematics, Benacerraf's identification problem is a philosophical argument developed by Paul Benacerraf against set-theoretic Platonism and published in 1965 in an article entitled "What Numbers Could Not Be". Historically, the work became a significant catalyst in motivating the deve…

Thanks, that was an interesting rabbit hole. Although I can't help but feel there's a philosophical map/territory confusion here. Like, sure, numbers can't possibly just "be" sets because there are many different models of the naturals in (ZF) set theory, even in higher order logics. But I feel like a Platonist would just counter that of course this is the case - we are simply modelling the properties of the "true" n…

Could we just say that natural numbers can be represented using sets, and leave it there?

Natural numbers can also be represented by even natural numbers (e.g. the easy way where 2n represents n), but that doesn't tempt people to make metaphysical statements about natural numbers somehow fundamentally "being" even. There is no reason why a representation of natural numbers by sets should be any more tempting.

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