Mindscape 282 | Joel David Hamkins on Puzzles of Reality and Infinity

The philosophy of mathematics would be so much easier if it weren't for infinity. The concept seems natural, but taking it seriously opens the door to counterintuitive results. As mathematician and philosopher Joel David Hamkins says in this conversation, when we say that the natural numbers are "0, 1, 2, 3, and so on," that "and so on" is hopelessly vague. We talk about different ways to think about the puzzles of infinity, how they might be resolved, and implications for mathematical realism.

Joel David Hamkins received his Ph.D. in mathematics from the University of California, Berkeley. He is currently the John Cardinal O'Hara Professor of Logic at the University of Notre Dame. He is a pioneer of the idea of the set theory multiverse. He is the top-rated user by reputation score on MathOverflow. He is currently working on The Book of Infinity, to be published by MIT Press.

#podcast #ideas #science #philosophy #culture

[FAAMS]

This is in my own opinion the best episode so far...nothing of what has been said is new but it is crystal clear for a good bunch of people who were not aware of it, or that didn't take it to seriously. Joel commitment to clarify is brilliant!

[mainsequence]

WOW, this is one guest I thought would be nice to have on here but I didn't actually expect it to happen. Thank you Sean!!

[futurisold]

What a pleasant surprise to see Mr. Hamkins here!

[DigitalGnosis]

Hey, Ive interviewed him! Love his book on proofs!

[jay]

This discussion about our universe was humbling even to Sean Carroll. Thank you for allowing us all to eavesdrop on this discussion.

[nda]

Couldn’t have been any better! Great conversation!

[ehfik]

marvellous

now can we get an episode with misha gromov, pretty please!

[josephrichards]

15:59
Damn this was so insightful!
It is so generalisable to different areas of thought.
As long as we don’t take this truth as foundational right?!;)

[charliesteiner]

This was great, I just wish it had been longer, and that the extra time had been used to thoroughly explain some of the ideas, like "model" and "large cardinal" that got used rapid-fire.

[rumidude]

What explains me sitting through and hour and eighteen minute discussion of mathematics which I understood almost nothing and yet stayed with it?
Regardless I it did get me to ponder a few things both old and new.

[Zirrad]

Godel's theorem limits one set of axioms from completely spanning the space of true statements in a formalism.
What prevents the use of multiple sets of axioms that, while incomplete themselves, fully (or more completely) span the space of true statements?
Edit: Ok, answered repeatedly later in the podcast...

[davidwright]

'A googleplex is an enormous number'. Surely, in terms of the cardinality, Alef(0), of the integers, it's tiny indeed compared with (googleplex upward!) many of its (numerically) distant successors! Also, notions of 'potential' infinity (in the sense of 'if you go far enough, but haven't yet done so'), the notion of time and mortality creeps into what's supposed to be a timeless, logical structure? Surely you 'have' all the positive integers given 0 (zero) and the successor function? In that sense, the integers are very compressible indeed! Of course, 'successor' implies an order.

[jayvincent]

Thank you... Need some Valuable content asap... Tired of force fed media today...

[uubuuh]

what is that riff adapted from? it’s on the tip of my tongue 4:33

[stephanecouvreur]

The phrasing “There are statements which are *true* but cannot be proved” should be avoided. It is confusing, because provability is well-defined in maths, but truth isn’t!

Better to say “There are statements which, if proved or disproved, would lead to a contradiction.”

Therefore, if we assume that there is no contradiction in our axioms, such statements can neither be proved nor disproved. This means we can add them-or their negation!-to the axioms without any contradiction. One famous example is the continuum hypothesis.

Asking to limit the use of the word "truth" in mathematics is a lost battle? Yes, I know! 🙃

[Hecarim]

You can CONSTRUCT Infinities but infinities aren't real ツ
==>
At least not yet
¯\_(ツ)_/¯

[ovidiulupu]

Existence of Planck constant mins that numbers are integer. Any other number îs a mental construct. So geometry must begin discret and în The limit of large numbers aproximate continuum. Time is discret, energy and mass are discret. În computers are discret proceses. Scale change, discret obvious, simulate continuum reality. I Began to think operations with numbers as operators. Identical independent particles made clasical mathematics posible. Enthengeled particles rise another kind of number, element of another kind of set.

[jjzrman]

Buy the definition of infinity. There really can't be more than one type of infinity.
I mean, is there more types of nothing? Can you have zero of something or less than zero of Something and still be zero...?.... I don't know help me out here It seems like some type of oxymoron