This Saturday at 2PM Eastern time, I’ll be talking to the Philadelphia Association for Critical Thinking on “Why is There Something Instead of Nothing?”
Unfortunately, times are such that I’ll have to give my talk over Zoom. Happily, this means that no matter where you are in the world, you can attend. Register by visiting the upcoming meetings page and scrolling down to “Click Here to Register” (near the very bottom), or just visit the registration page. Registration is free but required.
Bold topic. Go get ’em. Any slides or other hints available for preview?
antianticamper: Sorry; no previews. I expect to be tinkering with slides right up till the last minute, as is my wont.
So, by 2pm, maybe you’ll have decided that there’s really nothing after all?
‘Cuz, after the final season of Game of Thrones, I’d begun to suspect this….
Wait for the books! I am sure it will be all turn out much more satisfatory. Apparently Martin does not plan everything out, but lets the muse take him as he writes. I think the HBO series has deprived him of his muse.
Excellent lecture, but I wish there had been time to pose my question to Steven, which was: “How do we ‘falsify’ (a la Karl Popper) mathematical axioms?
F.+E.+Guerra-Pujol (#5):
One falsifies mathematical axioms by demonstrating that they are false. For example, supppose you propose the following axiom:
“Every even number greater than two is the sum of two primes”.
I can falsify this by writing down an even number, writing down every prime that is less than that number, adding those primes in pairs in every possible way, and showing that none of those sums is equal to the number I started with.
Thanks for the clarification, but I am unpersuaded for two reasons. Your first sentence states a tautalogy, but secondly and more importantly, I have always thought of axioms as propositions whose truth we simply take as a given (amd thus beyond the reach of falsification).
F.E. Guerra-Pujol: If your rule is that nothing you take as given can ever be falsified by any evidence, then you might have a good career in politics ahead of you.
That is not “my rule” at all (especially since I am a Bayesian); however, that is rule for mathematical “axioms,” is it not? Accordingly, I would change the word “politics” for “mathematics” in your reply!
F.E. Guerra-Pujol:
Will a simpler example help?
Axiom: Every odd number is prime.
Falsification: 9 is odd (because it is one more than 8, and 8 is even because 8 can be divided by two) and 9 is not prime (because 9=3 x 3).
If that doesn’t count as a falsifcation, then I guess I have no idea what “falsification” means to you.
I get both examples, but hear me out: if I postulate “all odd numbers are prime” as an axiom for an alternative mathematical system, then by deinition don’t we have to accept that statement as true for that particular mathematical system, i.e. an alternative system in which odd integers are prime?
F.E. Guerra-Pujol: I think we’re failing to communicate for the usual reason, which is that English words take on different connotations depending on context. I write down some axioms. I then look for mathematical structures that satisfy those axioms. For example, one of my axioms might be: “For all a and b, ab=ba”. This is true if a and b are integers. It is false if they are matrices. I say that the integers are a “model” for these axioms. Now:
Axioms, Take One. The goal is to write down a list of axioms that have many different models. This is good, because then every time we prove a theorem using our axioms, we’ll know that theorem is true in all of our models. We do the work once, and get to use the fruits of our labor many times. Of course, not every mathematical structure will be a model of our axioms, but we hope that many will. There is no issue of whether the axioms, on their own, are “true” or not — they are true statements about some structures and the same axioms are false statements about other structures. This, I think, is the sense in which you are talking about axioms. It’s sort of like thinking of an axiom as an “assumption for the sake of argument”. We make the assumptions, we draw a conclusion, and we trust our conclusion in any and all circumstances where we can verify that our axioms are true.
Axioms, Take Two But the above has essentially nothing to do with what Peano was trying to accomplish with his axioms for the natural numbers. Peano was not interested in broad theorems that could be applied to many structures — he was interested in one particular structure, namely the natural numbers —- and his goal was to write down a list of axioms that define the natural numbers, in the sense that only the natural numbers satisfy all the axioms. (In the jargon above, this means that the natural number are the only model of the Peano axioms.) In Axioms, Take One, the entire goal is to write down axioms with a great many models. In Axioms, Take Two, the entire goal is to write down axioms with exactly one model, namely the model you’re interested in studying — in this case the natural numbers. And here, it makes perfect sense to ask whether an axiom is true or not. If it’s true in the intended model, it’s true. If not, not. So for this kind of use of axioms, the axiom “Every odd number is prime” is just plain false. And it can easily be falsified by demonstrating that it is false.
Now Peano failed, because his axioms do in fact have alternative models, beyond the intended one (though those models are extremely weird — it is possible to prove, for example, that addition and multiplication in those models are so complicated that no computer can be programmed to perform them).
It turns out that having many models is equivalent to there being true statements in the intended model that your axioms can’t prove.
So we have:
Peano’s goal: Write down TRUE axioms that CHARACTERIZE the natural numbers, in the sense that the natural numbers are the ONLY model, which requires the axioms to be COMPLETE, in the sense that every true statement about the natural numbers can be derived from those axioms.
Godel’s quashing of that goal: The Peano axioms are not complete, and therefore do have other models. Likewise for absolutely any other system of axioms you could possibly write down.
In the talk you saw, I was referring to axioms in the “Take Two” sense.
If you poke around YouTube, you’ll find a talk I gave at the U of WIsconsin a few years ago which overlapped a lot with this week’s talk, but where I spent a fair amount of time talking about the difference between “Take One Axioms” and “Take Two Axioms”. I chose to eliminate that from this talk in the interest of not digressing too much, but I fear this was a disservice to thoughtful listeners like yourself.
I hope this helps.
Yes, this was very helpful. Thanks, I get it now!