The Deep Problems In Mathematics Are Being Ignored

Justin Clarke-Doane, professor of philosophy at Columbia University, talks about the foundations of maths, Zermelo-Fraenkel set theory, Axiom of Choice and logic.

#philosophy #maths #learn

[NoName-ylpb]

It is no coincidence that some of the greatest philosophers of the enlightenment were also mathematicians

[jacksonkohls]

It's extremely normal to take a course in set theory and building up things from axiomatic principles during a math major. Your correct that the exposure is usually limited (usually twice- set theory and real analysis), but knowledge about the axioms is more important than being able to rattle them off off the top of your head.

[ywtcc]

Comp Sci students are exposed to logic and set theory pretty early on.
It seems to me that a lot of pure math students are getting there through this Comp Sci route now.
Personally, I think that Comp Sci might be a misnomer. It's an extension of the study of Algorithms.
The mathematics and analysis coming from the study of Algorithms turns out to be exceptionally useful!
It looks like AI, philosophy, fundamental mathematics and science are convergent problems, and I'm hopeful there's progress around these issues in the near future.

[BenDjinn]

I think politicians and ceo's should be required to pass a logic course every 5 years

[Eta_Carinae__]

I think Paul Benacerraf sufficiently demonstrated that there was an underdetermination of foundations (at least in natural numbers). It's probably the case that foundations in general are underdetermined, and mathematics is rather a set of truths about much more general structures than even the proposed foundations - that funnily enough, the truths of math are more foundational than its foundations. I think this might be a large part of what motivates much of mathematics to gloss over foundations, if I were to bet on it.

[guidotoschi]

Here in Italy at the University of Genova in the faculty of Mathematics the course in Mathematical Logic is mandatory. Besides you can also take, if you chose, a course in Cathegory Theory and one in Axiomatic Set Theory.

[Stacee-jxyz]

1) Calculus Foundations

Contradictory:
Newtonian Fluxional Calculus
dx/dt = lim(Δx/Δt) as Δt->0

This expresses the derivative using the limiting ratio of finite differences Δx/Δt as Δt shrinks towards 0. However, the limit concept contains logical contradictions when extended to the infinitesimal scale.

Non-Contradictory:
Leibnizian Infinitesimal Calculus
dx = ɛ, where ɛ is an infinitesimal
dx/dt = ɛ/dt

Leibniz treated the differentials dx, dt as infinite "inassignable" infinitesimal increments ɛ, rather than limits of finite ratios - thus avoiding the paradoxes of vanishing quantities.

2) Foundations of Mathematics

Contradictory Paradoxes:
- Russell's Paradox, Burali-Forti Paradox
- Banach-Tarski "Pea Paradox"
- Other Set-Theoretic Pathologies

Non-Contradictory Possibilities:
Algebraic Homotopy ∞-Toposes
a ≃ b ⇐⇒ ∃n, Path[a, b] in ∞Grpd(n)
U: ∞Töpoi → ∞Grpds (univalent universes)

Reconceiving mathematical foundations as homotopy toposes structured by identifications in ∞-groupoids could resolve contradictions in an intrinsically coherent theory of "motive-like" objects/relations.

3) Foundational Paradoxes in Arithmetic

Contradictory:
- Russell's Paradox about sets/classes
- Berry's Paradox about definability
- Other set-theoretic pathologies

These paradoxes revealed fundamental inconsistencies in early naive attempts to formalize arithmetic foundations.

Non-Contradictory Possibility:
Homotopy Type Theory / Univalent Foundations
a ≃ b ⇐⇒ α : a =A b (Equivalence as paths in ∞-groupoids)
Arithmetic ≃ ∞-Topos(A) (Numbers as objects in higher toposes)

Representing arithmetic objects categorically as identifications in higher homotopy types and toposes avoids the self-referential paradoxes.

4) The Foundations of Arithmetic

Contradictory:
Peano's Axioms contain implicit circularity, while naive set theory axiomatizations lead to paradoxes like Russell's Paradox about the set of all sets that don't contain themselves.

Non-Contradictory Possibility:
Homotopy Type Theory / Univalent Foundations
N ≃ W∞-Grpd (Natural numbers as objects in ∞-groupoids)
S(n) ≃ n = n+1 (Successor is path identification)
Let Z ≃ Grpd[N, Π1(S1)] (Integers from N and winding paths)

Defining arithmetic objects categorically using homotopy theory and mapping into higher toposes avoids the self-referential paradoxes.

[joshnicholson]

Philosophy grad turned mathematics student. Yeah.

[vickydixon]

The axiom of replacement is speaking about rotations around a center axis it has a notation that looks like a 🍩 donut with an axis ↑ through it, the laws of physics remain the same, you are just expanding the set to account for what always existed, but you didn't know till then. You didn't find anything new, you just found out. That's the issue. It's already been answered that number set has existed for thousands of years, on the account that there have been billions of humans making these sets and measuring the Stars, you might even say that they were great great grandparents / our direct blood lineage that stored what they saw, heard, and felt in their part of the world, in their time, in books. For us. That's what books are for.

[hibald]

Just like there is no philosophy free physics, there also is no philosophy free math.

[martinwulf]

Anyone who’s studied maths at university has studied set theory, logic, axioms, and in particular what happens when you change the axioms.

No one is saying the axioms are true or not true. They’re what is assumed to be true. But you’re allowed to change your assumptions, and that changes the outcome of the logical deductions you use for your proofs.

[fishboy]

Ya I think the people that aren’t doing the philosophizing need to know what is happening in the philosophers department to know when they come onto a problem they think is new that that is what they signed up for when not doing the philosophy part.

[robsollart]

Does all of this mean that Leopold Kronecker was right when he reacted to Georg Cantors argument about infinite sets: I don't know what prevails here, Philosophy or theology, but it's not math.
( I know this from wikipedia, controversy over cantor's argument )

[jacksonbittner]

It's not normal? Logic is required for law school. Everyone I knew in college took it. Maybe it's not normal in stem?

[samueldeandrade]

Oh the "philosophical drama" in Math. Useless af.

[blake]

Logic is a pretty common course in math

[jameskelly]

Math and psychics community move on because the evidence for intelligent design makes that theory more and more compelling. Almost like philosophy and theology are the foundation for all knowledge. Almost like a certain religion has been saying we’re made in the image of god so we have the capacity to innovate and understand.

[onebigd]

You clearly don't know know what you're talking about