AskAMathematician.com presents a lecture on the foundations of math and whether we really can know that one plus one equals two. How was math invented? Where does mathematics come from? Are the axioms of math provable? Is math true? Can it be proven on purely logical grounds? Can it be demonstrated empirically? Can it only be justified from a pragmatic perspective? These are some of the questions discussed in the three videos below.
Part 1 of 3:
Part 2 of 3:
Part 3 of 3:
Tags: 1+1, Axioms, Foundations, Godel, Incompleteness, Logic, Math, Mathematics, Proof, Provable, Prove, Truth
Great video.
I recently got into a discussion on how the universe’s exclusion principle as described by Pauli is a foundation for mathematics: since electrons, protons and neutrons exhibit space-occupying behaviour, the common elements in our experience follow the axioms of set-theory (at least in the finite cases), and so math is applicable to the real-world. If there was no exclusion principle, then our math would be much different (assuming we were here to derive math at all).
Mathematician, I really enjoyed these. Surely there are more to come?
I do feel though that cramming so many topics together seems overly ambitious, as the connections between them feels weak.
The audio track needs to be normalized, by the way, the volume is kinda low.
For some reason I was under the impression that Bertrand Russell had taken care of the 1+1=2 problem. Wasn’t there three-hundred and something pages in the Principia Mathematica that proved it once and for all using set theory without the axioms? What else have they been lying to me about?
Okay, well Wikipedia says that he used the axioms of type theory plus three other axioms. I guess I was misinformed. Bah!
What’s the difference between “set theory” and “type theory?” It seems like you alluded to it in the video.
DFB: Set theory (or naive set theory) is vulnerable to paradoxes; the best known is Russell’s Paradox, the set of all sets that don’t contain themselves. Type theory was an attempt to remove these paradoxes by imposing a class hierarchy: sets could only contain objects of a “lesser” class.
The hope was that type theory would provide a solid (non-paradoxical) foundation for mathematics. That hope was dashed by Godel’s incompleteness results; however, type theory found another use in modern computer languages, so not all was lost…
I greatly enjoyed your talk
You know, I wonder if you could adapt this to a TED talk, I’m sure that audience would find this fascinating as well.
One old philosophical question that your talk reminded me of is the age old question, “Is mathematics discovered, or invented”? I personally believe that the answer is somewhere in between these two choices. In some sense, mathematics describes characteristics and behaviours inherent in the universe, so you could think of it as a “discovery”. But in another sense, it describes things that have no relationship with the natural world, it can be a human “invention”. I’d like to hear your take on it
I have heard of Godel’s incompleteness theorem before, but I had not heard this idea that math cannot answer “Is there a cardinal number between aleph-0 and aleph-1?” I’d love to hear more examples and descriptions of where the limit between what math can prove and express, and what it cannot.
I saw your video about how we know 1+1=2. Anyway, my philosophy on the subject is that mathematics is ultimately a system of tautologous statements. For instance, we know that one plus one equals two because two is defined as one and one, so the statement boils down to “one and one is the same as one and one”.
Believing such a statement only requires that we know that can believe tautologies. How can we know that A = A, that B = B, that one and one = one and one? My answer: because we directly experience the truth of these statements. By direct experience we can determine that things are the same as themselves. And that type of direct experience cannot be doubted, therefore it needs no defense.
What do you think?
If math is true only in the same sense that any set of tautologies is true, then it is a very trivial and uninteresting form of “truth”. In that case, “1+1=2″ is just a complicated way of saying “A is A.” I think that many people who claim that math is true mean something more than this. However, if you take the axioms as given, then it is indeed true that everything else follows just by definition and by application of the axioms.
Sounds to me that you two’s thoughts agree, that the axioms are devised & modeled on what we intuitively think of as true, and what follows are then logically (tautologically?) true. My wording is probably not very good though.
I’m sure I’ve heard many times in class when professors teach a new definition/theorem, it goes like this: “So we want to define XXX, and intuitively it should do YYY and ZZZ, so let’s assume YYY & ZZZ… It turns out that this works perfectly/is just what we want/etc. except…”