24 Responses to “Where I’ve Been”

1. 1 1 Harold
   October 1, 2020 at 9:45 am

   I think in the endless argument between the tortoise and Achilles, the tortoise should have met Achilles half way.

   I was jotting down ideas as I go, so many of the things are addressed in your presentation and are just ramblings.

   Why is there something? “maybe there is no answer” is not the only possibility.
   We could perhaps demonstrate in principle that “nothing” is impossible.

   The argument reminds me of the Ontological argument. I am not at all saying it is the same, but has a similar “flavor” in discussing the necessity of existence. This is something you refer to with the comparison with the religious arguments.

   The subject of existence is horribly complicated. The conflict between possibilitists and actualists is an example. The actualist proposition is:
   “everything that exists is actual” or “There is nothing that is not actual”. This sort of fits your unicorn example. The blue stripe is undeterminable because the unicorn is not actual.

   This apparently intuitive proposition leads down a rabbit hole of complexity that is contradicted by SQML (Simplest Quantified Modal Logic). This insists that things that could possibly exist do exist. There are get-arounds, but they are open to reasonable criticism.

   Upshot: if we don’t like the actualist get-arounds, then there exists a possible unicorn. Not that unicorns could possibly exists, but there exists a possible unicorn. This contradicts the actualist seemingly sensible proposition that everything that exists is actual. The possible unicorn is not actual in our universe but it exists. I think you agree with this, as all possible universes do exist in your view. Going back to your first point about unicorns, the statement every unicorn has a blue stripe is true if it possible that such a universe exists.

   I am not sure where that leads us. We could conclude that philosophical existence is different from what we generally refer to as existence. The Godel universe exists, but so do possible unicorns in perhaps the same way. Nevertheless, unicorns are not actual in our universe. Alternatively, we could conclude that the claim that mathematical objects exist is not at all controversial.

   I guess you address this with the corn eating universe. Unicorns do not exist in our universe, but they do exist if they are mathematically possible.

   I am not sure that comparing your hypothesis with a discredited religious hypothesis is such a good way to conclude. I dismiss the religious guys “intellectual fulfillment” as mistaken, or at least not one that can extend reasonable to anybody else. I think your arguments are way better than the religious ones, which you go on to explain, but saying it is better than a rubbish one does not seem such a great way to finish.

2. 2 2 Harold
   October 1, 2020 at 9:48 am

   OMG! Everything has gone orange! Is it just me, a new style, the fires, or some awful spreading from the orange man?

3. 3 3 Daniel R. Grayson
   October 1, 2020 at 8:22 pm

   If you say that something “exists” if the propositions (or properties) about it are either true or false, then how do you define “proposition”? Most people would say that a proposition is a statement that is either true or false.

   Moreover, it’s hard to tell whether a statement is “about” unicorns. Take this one: “for every unicorn, 2+2=4”. It seems to be about unicorns, but not “really”, and it seems to be true.

   Thirdly, some things don’t exist, but there are propositions about them anyway. Example: every composite prime number is prime. (This is very much like saying every unicorn has a blue stripe: it is true, trivially, because there are no unicorns.)

4. 4 4 Roger
   October 1, 2020 at 11:10 pm

   At 34:00, you describe mathematical truth.
   At 35:00, you describe mathematical Platonism.
   At 40:30, you say physical theories are approximations to truth.

   I can accept all of that, but then at 40:50 you say:
   “The one hypothesis that underlies every viable physical theory is that the Universe is a mathematical object.”

   Here is where I differ. I don’t think that our Universe is a mathematical object, or that any of our theories assume that it is. Our physical theories are mathematical approximations to a non-mathematical object.

   This puts me at the opposite extreme from Tegmark’s hypothesis. While he says all mathematical objects are universes, I say that none of them are.

5. 5 5 Steve Landsburg
   October 4, 2020 at 11:29 am

   Daniel R Grayson (#3): This is what I get for translating my somewhat imprecise ideas into even more imprecise ideas for the purpose of talks like this. You deserve a good answer, and I’m going to try to find time to write one up for you.

6. 6 6 Steve Landsburg
   October 4, 2020 at 11:35 am

   Roger:

   Here is where I differ. I don’t think that our Universe is a mathematical object, or that any of our theories assume that it is. Our physical theories are mathematical approximations to a non-mathematical object.

   I think, then, that you and I agree that physical theories are almost always purely mathematical, and that they are only approximations to reality. Where we differ is that I expect the reality they’re approximating is also purely mathematical, and you think it’s something else.

   But I don’t see why you would think that. First, if every approximation we can come up with shares a single fundamental characteristic, that seems to me like a pretty good clue that the thing we’re approximating has that characteristic.

   But second, and more importantly, I can’t even begin to imagine what that “something else” might be, or what is gained by thinking that there’s a “something else” that we seem to have no hope of understanding.

7. 7 7 Roger
   October 4, 2020 at 4:17 pm

   You say that our physical theories are mathematical, and you cannot imagine what else they could be. Okay, but that is no evidence that the Universe is a mathematical object. Only that we only know how to do approximations mathematically.

   We don’t really even have any candidate for the Universe as a mathematical object. There are those who talk about having a quantum wave function of the Universe, but they end up talking about many-worlds and other ideas that have never made any sense or had any predictive value.

8. 8 8 Daniel R. Grayson
   October 7, 2020 at 7:48 am

   Roger and Steve,

   Re: “that is no evidence that the Universe is a mathematical object”

   What would such ‘evidence’ be? Unless we can imagine an experiment whose result would shed light on whether the universe is a mathematical object, the statement that it is one is undefined and must be regarded as non-physical.

   Let me tell a related story. As a mathematician I am used to imagining that the objects of mathematics exist — after all, I study them. In particular, I imagine that the “sets” of “set theory”, upon which mathematics is traditionally based, exist. In 2011-12 I encountered something novel — a new logical system upon which mathematics can be based: “type theory” (or more precisely, “homotopy type theory” and “univalent foundations”). Type theory comes with software which will accept statements and proofs formulated in it and will verify their correctness. Interacting closely with such software and creating some of my own gave me the strong impression that this way of doing mathematics was just a game with strings of symbols and with precise rules, rather than a way to view of a new mathematical world with novel inhabitants. It caused me to feel some sort of deep distress. It took me about 3 months to shake that feeling and regard the new world and its inhabitants as “existing”, and after that I felt much more comfortable. Indeed, now I regard those who think just about sets as missing out on the larger world of types. It’s a more fun world to explore.

   To return to Steve’s basic thesis, that mathematical objects exist because they have properties that we can study: that *is* the sort of existence that matters to mathematicians. We can prove things about sets and types, just as though they existed, so why not regard them as existing? It doesn’t hurt, and it adds a degree of comfort. But try to give a definition of “existing”, and we’ll fail. It really is just a game with symbols and precise rules.

   Maybe physics is the same: once we have a complete mathematical theory of the universe, will physicists derive some comfort from imagining that the theory is all there is? Or will they insist that there is something more, without being able to define what it might be?

9. 9 9 Steve Landsburg
   October 7, 2020 at 1:17 pm

   Daniel R. Grayson:

   1) I’ve been fascinated by type theory since I read this paper, which led me to further reading and a little bit of playing around with Coq.

   2) As far as the “existence” of mathematical objects, I think some objects are on much firmer grounds than others. I very much doubt that there is anything that deserves to be called the universe of sets, and I didn’t need type theory to instill this doubt. I’m at least unsure about whether there is such a thing as a universe of sets (i.e. a model of ZFC), independent of the strings of symbols we write down and the rules we use to manipulate them. But that doesn’t require me to doubt the existence of the natural numbers.

   3) The natural numbers seem to me to have a clearer ontological status for several reasons. First, they are absolute — if you construct the natural numbers in two different models of ZFC, you’ll get exactly the same natural numbers both times. (I realize this invokes models of ZFC, of which I’ve already expressed skepticism, but I believe this argument can be reformulated to avoid that problem.) Second, unlike sets, mathematicians had absolutely no problem reasoning about natural numbers long before anyone thought of axiomatizing them. Third, a certain amount of knowledge of the natural numbers seems to me to be a *prerequisite* for type theory (or anything else): The concept of a finite string of symbols, and the distinction between the first, second and third strings in an argument, are part of arithmetic.

   4) I understand you as suggesting that it really doesn’t matter whether things “exist” in some absolute sense, and doesn’t even really matter whether we have a good definition of what “existence” means; existence, for all practical purposes, is just a statement about which thinking habits we find comfortable. I have great sympathy for that (which does not erase my obligation to come back and tighten up my proposed definition as promised — but not yet delivered — in my response to your previous comment). In that way of thinking, my point is that nothing is gained by treating physics any differently than any other branch of mathematics in this respect.

10. 10 10 Daniel R. Grayson
    October 8, 2020 at 11:33 am

    Re: “I’m at least unsure about whether there is such a thing as a universe of sets (i.e. a model of ZFC),”

    In the approach of von Neumann and Bernays to extending ZFC, they introduce a new type of thing called a “class”, and there is a class of all sets. Moreover, their system is conservative — no new theorems can be proved. One could imagine then introducing a new type of thing called a “hyperclass”, with a hyperclass of all classes. And so on. Perhaps such a system is also conservative over ZFC. This would look much the same as the ascending sequence of universes U_0, U_1, U_2, …, encountered in type theory and in Coq.

    I wonder whether that is a way to bypass the impulse to ask for a model of ZFC within ZFC.

11. 11 11 Richard D.
    October 9, 2020 at 6:24 pm

    So, mathematical objects exist, in their own reality, independent of the human mind. And assertions in that domain are factually true or false, in the same way as “There was .2 inches of rain in Seattle last Wednesday.”

    Then, is the continuum hypothesis, regarding the real numbers, objectively true or false?

12. 12 12 Steve Landsburg
    October 9, 2020 at 6:33 pm

    Richard D. (#11):

    So, sheep exist, in their own reality, independent of the human mind. And assertions in that domain are factually true or false, in the same was “There was .2 inches of rain in Seattle last Wednesday.”

    Then, is the statement “it is black”, regarding a sheep, objectively true or false?

    Answer: It depends on which sheep you’re talking about. And this dependence is not a reason to doubt the objective existence of sheep.

    Your final question is “Is CH, regarding the real numbers, objectively true or false?”. As stated, the question makes no sense, because CH is not a statement about the real numbers; it is statement about models of ZFC. So I will assume this is a typo for “Is CH, regarding a model of ZFC, true or false?” Answer: It depends on which model you’re talking about. And this dependence is not a reason to doubt the objective existence of models of ZFC.

13. 13 13 Steve Landsburg
    October 9, 2020 at 10:11 pm

    Daniel R Grayson (#10): If I understand your vision, it’s essentially the same as positing the existence of arbitrarily large Grothendieck universes (i.e. “for every cardinal, there exists a Grothendieck universe of higher cardinality than the given cardinal”). Or have I missed your point?

14. 14 14 Steve Landsburg
    October 9, 2020 at 10:25 pm

    Daniel R Grayson (#3):

    some things don’t exist, but there are propositions about them anyway. Example: every composite prime number is prime. (This is very much like saying every unicorn has a blue stripe: it is true, trivially, because there are no unicorns.)

    This is certainly true according to the standard logic of mathematics. But philosophers who think about ontology — including some quite mathematically sophisticated philosophers like Hilary Putnam — have tended to the view that if there are no boojums, then statements of the form “Every boojum is a snark” should be regarded as meaningless, not as true.

    That’s the language I adopted when I said that it is neither objectively true nor objectively false that every unicorn has a blue stripe.

    In other words, the following two statements have the same meaning:

    A) The statement “Every unicorn has a blue stripe” is meaningless

    B) There are no unicorns.

    Now you might object that I’m out to prove B), and I’ve done it by asserting A), which is equivalent to B) by definition, so I’ve cheated you. But my purpose here is not to prove B), it is to establish the meaning of B), and for that, I think A) is extremely useful.

    I have much more to say on this, some of it well thought out (or presumed well thought out until proven otherwise) and some of it sketchier. I hope to post more soon.

15. 15 15 Roger
    October 9, 2020 at 11:06 pm

    Steve, I thought that you were a mathematical Platonist, until I read #12.

    CH is very much a statement about real numbers. It says that every subset of the reals must be either countable or in a ono-to-one correspondence with the reals. If the reals exist objectively, and we knew everything about them, then surely we could answer this question.

    If someone asks you for truth, and you evade the question by arguing that it depends on the axioms, then you are taking the stance of the anti-platonists. They regard the reals as just formal inventions that don’t necessarily have anything to with reality.

16. 16 16 Steve Landsburg
    October 9, 2020 at 11:19 pm

    Roger: I am a platonist regarding the natural numbers and an agnostic leaning toward disbelief regarding the reals.

    Also, CH is a statement about arbitrary subsets of the real numbers. That is not the same thing as a statement about the real numbers. Likewise, statements that are really just about the real numbers are essentially statements about arbitrary subsets of the natural numbers. Those are certainly not the same thing as statements about the natural numbers.

    Also, one needs to be careful about what one means re being a platonist about the real numbers. There is platonism with respect to the assertion that there exists some model of the real numbers, and there is platonism with respect to the assertion that there exists a preferred model of the real numbers. These are not equivalent assertions. (This distinction does not come up with regard to the natural numbers, because the natural numbers are absolute — they don’t depend on your model of ZFC, whereas the real numbers do.)

17. 17 17 Daniel R. Grayson
    October 10, 2020 at 1:26 pm

    Steve,

    Re: “In other words, the following two statements have the same meaning:

    A) The statement “Every unicorn has a blue stripe” is meaningless

    B) There are no unicorns.”

    I think that’s backwards: whether a statement is well-formed and has a meaning should be easy to determine, a matter of checking for definitions and proper syntax, not a matter of discovering further facts about the universe, such as whether unicorns exist.
    To give a mathematical analogue, the following statement should be regarded as
    meaningful: “every even natural number greater than 2 that is not the sum of two primes is a multiple of 577”.

18. 18 18 Steve Landsburg
    October 11, 2020 at 8:43 am

    Daniel R. Grayson (#17):

    whether a statement is well-formed and has a meaning should be easy to determine, a matter of checking for definitions and proper syntax, not a matter of discovering further facts about the universe, such as whether unicorns exist.

    I don’t understand what “should” means here. Whether a statement is well-formed and has a meaning is a matter of convention. Mathematicians have universally adopted one convention, namely the one you’ve summarized here. Philosophers, particularly ontologists, have often found other conventions to be more convenient.

    Consider this:

    If all crows are black and all black things absorb light, then all crows absorb light. (*)

    By the conventions of mathematics, this is true, and you don’t have to know what a crow is to determine that it’s true. By the standards of ontology, Hilary Putnam says this:

    In my view, logic, as such, does not tell us that (9) is true: to know that (*) is true I have to use my knowledge of the logical principle (A), plus my knowledge of the fact that the predicates “x is a crow”, “x is black” and “x absorbs light” are each true of just the things in a certain class, namely the class of crows, (respectively) the class of black things, (respectively) the class of things which absorb light.

    (There is much more detail in Putnam’s short book on The Philosophy of Logic.)

    Presumably the conventions have evolved differently in different disciplines because different conventions are convenient for different disciplines. I think it therefore makes sense for the same person, when discussing mathematics, to use the conventions of mathematics, and, when discussing ontology, to use the conventions of ontology.

19. 19 19 Daniel R. Grayson
    October 11, 2020 at 9:21 am

    Steve,

    Re: “Daniel R Grayson (#10): If I understand your vision, it’s essentially the same as positing the existence of arbitrarily large Grothendieck universes (i.e. “for every cardinal, there exists a Grothendieck universe of higher cardinality than the given cardinal”). Or have I missed your point?”

    Well, that wasn’t my point, but I think I was forgetting something, anyway, so it wasn’t a good point.

20. 20 20 Daniel R. Grayson
    October 11, 2020 at 2:47 pm

    Steve, Putnam’s statement that we need to know that “x is a crow” is true just when x is a member of a certain class is exactly the same as saying that we need to know that “x is a crow” is a well-defined predicate (for then you can define the class to consist of the objects x such that x is a crow). After the passage you cite, he goes on to say that “x is a crow” is a pretty well-defined predicate, so that lends support for my view.

    I don’t see any place in Putnam where he insists that such classes should be nonempty.

    In particular, if we define “x is a unicorn” to mean “x is a horse with a horn emerging from its forehead”, then Putnam would agree that “every unicorn has a blue stripe” is a meaningful statement and that it’s false.

    So, as far as I can see, Putnam and I have the same conventions.

21. 21 21 Steve Landsburg
    October 11, 2020 at 5:41 pm

    Dan: I pulled the quote without re-reading the context, which I thought I’d remembered well. I’ll go back and re-read before responding.

22. 22 22 Daniel R. Grayson
    October 11, 2020 at 8:33 pm

    “and that it’s false.”

    Typo: I meant “and that it’s true (vacuously).”

23. 23 23 Roger
    October 12, 2020 at 2:44 pm

    Steve, I think you are deviating from mainstream mathematician beliefs. I think that most would have a platonist belief in the integers and the reals.

    There is a theorem that every nonempty bounded set of reals has a least upper bound. I would have said that is a statement about the real numbers, but I guess you disagree, and say it is a theorem about subsets of reals.

    “because the natural numbers are absolute” — I don’t get this distinction. If you are relying on axioms for the natural numbers, then you are going to get nonstandard models. The natural numbers will look strange in some models.

    If you are not sure whether you believe in the reality of the real numbers, then I don’t see how you ever get to the universe being the embodiment of some mathematical object. You ask what else it could be? Okay, I find it hard to imagine time being anything but a real number. But what does this say, if a real number is just some formal construct in a variety of possible ZFC models?

    To get to a universe, you need to represent fields and wave functions and other complexities, but these are even harder to grasp if we don’t agree on what a real number is.

24. 24 24 Daniel R. Grayson
    October 14, 2020 at 2:49 pm

    Roger and Steve, perhaps we should consider the possibility of believing in the “existence” of mathematical objects only if they can be constructed intuitionistically, without the use of axioms (such as the law of excluded middle or the axiom of choice). The real numbers would exist, but nonstandard models wouldn’t. Moreover, if we think of the universe as a simulation obeying certain rules, there’d better be an algorithm for it to execute.