• Arguing About Mathematical Objects

     

    William Lane Craig and James Sinclair’s chapter on the Kalam Cosmological Argument in the Blackwell Companion to Natural Theology contains a couple of arguments against the possibility of an infinite past.  Here’s one of them:

    P1.  An actual infinite cannot exist.

    P2.  An infinite temporal regress of events is an actual infinite.

    P3.  Therefore, an infinite temporal regress of events cannot exist.

    I’ve already talked about this argument in a few places:

    Infinite dreams

    Infinity minus infinity

    Can God count to infinity?

    And a few more posts will be devoted to this topic.  But here I’ll consider another angle.

    In defending their argument, Craig and Sinclair consider several possible refutations to the premise P1.  One of these is that the existence of infinitely many numbers (and other kinds of mathematical objects) might directly contradict P1.

    As Craig and Sinclair observe, this raises a very interesting question.  In what sense can it be said that mathematical objects like numbers exist?  Do they exist as material objects, perhaps only as configurations of neurons in the brains of humans?  Do they have some kind of disembodied existence in a platonic mathematical realm?  Or is there really no meaningful sense in which numbers can be said to exist?  Is there some other explanation?

    As a mathematician, I find these questions fascinating, though I certainly don’t claim to know the answers.  I’ve talked about these ideas with several of my colleagues over the years, and discovered that there are a range of views, and also a range of associated confidence levels.  The following diagram from the chapter illustrates several different kinds of theories of mathematical ontology (ie, theories about the existence of mathematical objects):

    mathematical objects

    There are certainly a lot of views!  To me, none of them seem likely to be provable – at least not in a strict sense – and I think the range of views held by professional mathematicians and philosophers counts as evidence that there does not currently exist a definitive proof one way or another.  With that said, though, I do have doubts about a few of them.  Some people have argued that we shouldn’t accept anything like a platonist view, for example, since there is no evidence that platonic objects exist.  But, according to a recent survey, 39.3% of professional philosophers are platonists (with 37.7% nominalists and 23.0% choosing some other option).  Most arguments I’ve heard seem to suggest that agnosticism is a reasonable position to take on the matter.

    Now, returning to Craig and Sinclair’s argument, we see that consideration of mathematical objects only proves to be a decisive refutation to P1 given certain realist views of mathematical ontology.  Here is how Craig and Sinclair put it:

    “The Realist, then, if he is to maintain that mathematical objects furnish a decisive counterexample to the denial of the existence of the actual infinite, must provide some overriding argument for the reality of mathematical objects, as well as rebutting defeaters of all the alternatives consistent with classical mathematics – a task whose prospects for success are dim, indeed. It is therefore open to the [apologist] to hold that while the actual infinite is a fruitful and consistent concept within the postulated universe of discourse, it cannot be transposed into the real world.”

    [As an aside, it seems that Craig and Sinclair are urging an insider of the Mathematical Realist camp to rationally assess his beliefs, and subject them to the same level of scrutiny he might demand of others.  Is this an Outsider Test for Mathematical Ontology?]

    The astute reader will have noticed that Craig and Sinclair are attempting to shift their burden of proof here.  They are making the argument.  And they must offer support for their premises.  If they are aware that the existence of mathematical objects could cast doubt on one of their key premises, then they must adequately deal with this possibility.  It is not enough to simply say that they think it would be difficult for an opponent to conclusively refute the premise.

    It is true that to conclusively refute P1 by considerations of mathematical objects, one must demonstrate the validity of a theory of mathematical ontology that involves the actual existence of mathematical objects.

    But, on the other hand, to conclusively prove P1, Craig and Sinclair (or someone else) must demonstrate the validity of a theory of mathematical ontology that involves the actual non-existence of mathematical objects – which I think Craig and Sinclair would probably agree is also “a task whose prospects for success are dim, indeed”.  If they do not successfully do this, we are left wondering if perhaps mathematical objects do provide a counterexample to P1.  But how might one prove such an assertion?  Proving the non-existence of mathematical objects seems akin to proving the non-existence of god(s)!

    In conclusion, I don’t know which theory of mathematical ontology is the correct one, and I don’t think Craig and Sinclair do either.  If some form of platonism is correct (as is believed by nearly 40% of professional philosophers), mathematical objects would indeed provide a decisive counterexample to P1.  And, unless this possibility can be ruled out, we have no compulsion to accept the premise – or the conclusion of any argument that relies on it.

    Category: Cosmological argumentInfinityLogicMathematicsWilliam Lane Craig

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    Article by: Reasonably Faithless

    Mathematician and former Christian
    • brad lencioni

      Well done!

    • DRC

      Great post about a fascinating topic.

      I would disagree on one point… The actual existence of mathematical objects would not necessarily be a counterexample to P1 – you would need the actual existence of an *infinite* mathematical object. It’s possible that the platonic realm contains just 2 types of objects: “1” and “0”, but not infinity. Computer scientists know you can do an awful lot with just 1 and 0. Since there is absolutely no evidence for the platonic realm, this binary system seems equally likely as every other unfalsifiable claim about this mystical place.

      My position is that there is no good reason to believe in the platonic realm, it adds no value to our understanding, and other more rigorous positions do a fine job to explain the existence of mathematical “truths”. I’ve never understood why Plato’s mere speculation has been elevated to such dizzying heights.

      • Reasonably Faithless

        True – it depends on how many mathematical objects exist. If 1 exists, then does 1+1, etc? Or do you just mean the symbols 0 and 1? Or do you mean Z_2 exists, so that 1+1=0? If some of the basic ways to combine sets (unions, power sets, etc) apply to existent objects, then all you need is the empty set in order to build up an enormous supply of mathematical objects. And the question of whether the empty set exists seems very interesting to me!

        I also haven’t seen any good reason to believe in the platonic realm, but neither have I seen a good reason to *believe* there is no such thing. Agnosticism seems OK to me on this one. I could happily call my self an a-platonist in the same way I call myself an atheist even though I don’t actively believe there is no god.

        And Craig’s argument requires a specific rejection of platonism, so it seems he needs to provide an argument to that effect, rather than demand critics provide an argument to prove a contrary position.

        • DRC

          Those are good questions about what “exists” really means for the platonic realm, and we have no idea because we have never seen this place. For me, the question is: What value does the concept of the platonic realm bring to us? Does it narrow down the types of mathematical proofs which are possible? Does it constrain any of the logical arguments which are consistent? Certainly not. The concept has none of the rigid conclusions that mathematics possesses. I could postulate that whenever we use the number 4, we’re actually referring to 4 unicorns in the platonic realm. Since every speculation about platonism is equally unsupported by reason or evidence, the concept is of no use – whether it’s true or not.

          • Reasonably Faithless

            But 4 equals { {} , {{}} , {{},{{}}} , {{},{{}},{{},{{}}}} } – no unicorns there! Although any collection of 4 unicorns could be put into one-one correspondence with the above set…

            But I think you’re right that thinking about the platonic realm does not alter the way we actually do mathematics. I guess it’s like the “I have no need for that hypothesis” statement attributed to Laplace.

            And I agree that any attempt to justify platonism will start to sound theological – eg, “of course you can’t *see* this place” 🙂

            • DRC

              Ah ha! But the empty set was invented by those unicorns – it’s an empty Vegemite jar which every unicorn carries on its back!

              But seriously… I think we can learn much about ourselves by asking why anyone believes in platonism (despite the lack of evidence). Perhaps it’s because when we manipulate formulas, we imagine the pronumerals have a life of their own, without referring to any physical object. We have evolved to think of objects as physical “things”, and information as “mental stuff”, so we make the leap to believing that information and thoughts must exist outside of physical reality. That was an old, but persistent misconception. We now know that all information is physical, and all thoughts are information processing, and that’s what mathematics is – the consistent manipulation of information. Platonism seems to be widely believed due to the false dichotomy of mind and matter.

              For platonism to be argued for, it really requires a defence that mind and matter are separate (despite the overwhelming evidence from neuroscience that the brain *is* a mind).

            • Reasonably Faithless

              But there are several empty vegemite jars! (Since there are several unicorns.)

              Yep, I definitely get what you’re saying, but it’s very hard to shake the feeling that if one spent one’s entire life counting, then there will still more numbers “out there” that just missed out on getting counted – being just as real as the ones whose names were spoken. Things like “We now know that all information is physical” are still assertions, essentially saying that platonism is false. I suspect that it would be possible to construct a theory of mathematics in which platonism plays absolutely no role whatsoever, and which describes everything we normally do with mathematics, so that platonism is an unnecessary assumption, and therefore unprovable. But platonism also seems to be a valid way to describe mathematics. So I see there being a bit of a stand-off, at least in terms of what can be proved. Perhaps by the same standard of evidence as most atheists apply to the god question, it is better to have a non-platonic view by default, until evidence is provided for platonism.

            • DRC

              “it’s very hard to shake the feeling that if one spent one’s entire life counting, then there will still more numbers “out there” that just missed out on getting counted”

              I don’t favour this argument. It’s like saying that if one has seen roses which are red, orange, and yellow, there *must* exist a green one, simply because we can imagine it. If a human being feels that something should exist, I don’t find this a compelling reason to think that it truly does. Keep in mind that to count to an enormous number, you need to store that number somehow, which takes some physical space and energy. From that point of view, it seems reasonable that there may be a largest number, above which it is not possible to count (somehow related to the size of the observable universe and available energy). I’m sure many mathematicians would balk at the idea that numbers are somehow limited by space, time and energy, but without those three things, it’s impossible to do any maths at all!

              Also, if the platonic realm exists and contains “all” the numbers, it must also contain the total sets of everything else: e.g. the name of Kim Kardashian’s nine-thousandth husband – if she were to have that many. It also contains a recipe which tastes “nothing like chicken”. In my opinion, the platonic realm is really a metaphor for all the things we can imagine are possible. Our minds are particularly adept at imagining things which don’t necessarily exist.

              I should clarify about “all information is physical”. I meant that all of the information *we have access to* is physical. It’s possible that non-physical information exists in a platonic realm, but then it can’t interact with our universe, so we won’t ever have that information. This is because the laws of physics describe a causally-closed system. You can call the statement “all information is physical” an assertion, but I would call it a very rigorous and falsifiable theory which could be disproven by providing just 1 example of non-physical information.

              “But platonism also seems to be a valid way to describe mathematics.”
              I’m not sure what you mean by this? What does platonism describe about mathematics, exactly? To me, platonism sounds like the vaguest notion possible, taking great care not to describe anything at all. The only specific description is that it’s non-tangible (to explain the total lack of evidence).

              BTW, the unicorns have agreed that it is immoral to count the vegemite jars as it would promote greed and jealousy. Officially, there is only “one” empty set because each unicorn is “one” with its own jar. 🙂

            • Reasonably Faithless

              “I don’t favour this argument”

              Actually, it’s not meant to be an argument for platonism, just an attempt to explain why people might assume it. Still, you’ve said some interesting things, so let me respond! (Please excuse a long-winded post…)

              “It’s like saying that if one has seen roses which are red, orange, and yellow, there *must* exist a green one, simply because we can imagine it.”

              I’d say it’s more like every time someone says “there isn’t a X coloured rose”, you end up finding an X coloured rose!

              “If a human being feels that something should exist, I don’t find this a compelling reason to think that it truly does.”

              Definitely agree. But I guess it’s more like saying that you feel like something should exist because there is a pattern. If you saw a pink rose, a slightly darker pink rose, another rose slightly darker than that one, etc, etc, etc, then you might reasonably suppose that you could find an even darker one than the darkest one you’ve seen yet. Bad analogy because there is a limit to rose darkness (as these are *definitely* made of matter). But, with numbers, the way you find a bigger number is to add 1 to the biggest one you know. Our lives are finite, so of course there is a limit to how high any one of us to count. But to say that *that* is the biggest number is to deny that someone who lives longer than you couldn’t say the next number. The human race (and all intelligent life) will eventually die out, but to say that the highest number is just the highest one counted (or used) by intelligent life is perhaps to say that it is logically impossible for an intelligent creature to have counted higher in another possible world.

              Also, I wonder what you’d think about this in the context of a multiverse. Suppose the greatest number ever used or counted by an intelligent creature in our universe was N. But suppose a being in another universe counted to a higher number M. Would you say that M has any kind of existence/meaning in our universe? What if there were P universes where P was a number much bigger than N? What if there are infinitely many universes, but each is finite in extent, etc?

              “I’m sure many mathematicians would balk at the idea that numbers are somehow limited by space, time and energy, but without those three things, it’s impossible to do any maths at all!”

              I think we’re all pretty comfortable with the fact that the mathematics we can do is limited by time and space. But I’m guessing that most mathematicians I know would think of that as there being more theorems that never got proved!

              “Also, if the platonic realm exists and contains “all” the numbers, it must also contain the total sets of everything else: e.g. the name of Kim Kardashian’s nine-thousandth husband – if she were to have that many. It also contains a recipe which tastes “nothing like chicken”. ”

              Really? I thought you made a good point a few comments back when you said the platonic realm might contain 0 and 1 but nothing else. How do you know it might not just contain the numbers, but not things like a perfect horse, or a recipe that tastes “nothing like chicken”?

              “In my opinion, the platonic realm is really a metaphor for all the things we can imagine are possible.”

              That might be right, yes. But you never know.

              “Our minds are particularly adept at imagining things which don’t necessarily exist.”

              Absolutely, but it’s also good at imagining things that do exist (and often before we knew they did).

              “I should clarify about “all information is physical”. I meant that all of the information *we have access to* is physical. It’s possible that non-physical information exists in a platonic realm, but then it can’t interact with our universe, so we won’t ever have that information. ”

              I think this is kind of what I’m thinking of. If there is a platonic realm, then there would be infinitely many theorems that *could* be proved, even though we won’t prove them all – and even though the shortest proof of some of them could not be written even if all the matter in the universe were converted to pens and paper. But just because we couldn’t access that information doesn’t mean it doesn’t exist – unless you define “exist” to involve interaction with our universe. Perhaps this information is available to beings in other universes – perhaps for each mathematical object, there is at least one universe in which that object can be directly apprehended? I know at least one philosopher who thinks that “reality” is maximally general – he has an interesting probabilistic argument for this.

              “You can call the statement “all information is physical” an assertion, but I would call it a very rigorous and falsifiable theory which could be disproven by providing just 1 example of non-physical information.”

              I don’t know how non-physical information could be provided. If the platonic realm is as it is usually described, then I don’t think it would be possible to interact with it in the sense of seeing, hearing, tasting, touching, or smelling something from it. So I think the claims “there is a platonic realm” and “there is no platonic realm” are equally unfalsifiable. (Just as the claims “there is a disembodied mind” and “there is no disembodied mind”.)

              “”But platonism also seems to be a valid way to describe mathematics.”

              I’m not sure what you mean by this? ”

              Nothing very concrete – just that somehow numbers, etc, exist “somewhere”, and we are doing mathematics, we are discovering stuff about numbers. I agree it’s vague, but perhaps someone has tried to make more sense out of this. I’ll try and dig something up!

              “The only specific description is that it’s non-tangible (to explain the total lack of evidence).”

              I don’t think the non-tangible aspect is specifically to explain a lack of evidence, but rather that that is what it seems like intuitively.

              “BTW, the unicorns have agreed that …”

              Finally some sense!! 😉

              Now, everything I’ve said above is mostly just playing devil’s advocate. I’m trying to say what might be said in defense of platonism. I doubt it could be proved, but I suspect it could stand up to any attacks.

            • DRC

              Fascinating discussion!

              “I guess it’s more like saying that you feel like something should exist because there is a pattern.”
              Yes, I think this is one of the key issues. It’s estimated there are ~10^80 baryons in the observable universe, so let’s pick a number much bigger than that. x=10^1000^1000. Now it seems we know a lot about this number, even if there’s nothing for it to physically represent. For example we can multiply it by 2…
              2*x=2*10^1000^1000

              This ability is a result of a simple pattern which emerges from our chosen axioms (just put a 2 at the front). I want to suggest that our human ability to follow this pattern does not imply the existence of x. Imagine another number y that’s a similar size to x, but is a long string of random digits instead of a one followed by many zeros. It’s impossible for us to calculate 2*y. The pattern is not as simple as for x, as it would require memory and energy far above what’s available to us. My point is that the ability to represent 2*x makes x feel very real, but we’re not really dealing with x as an existing object – we’re merely following a pattern to put a 2 at the front. It’s easy for humans to devise all sorts of patterns which don’t refer to any existing object e.g. the infinite repeating pattern: {Leprechaun, Fairy, Leprechaun, Fairy, Leprechaun…}

              To me, numbers seem to be an *information process* (or algorithm) rather than a thing. The number 3 is not an existing object in it’s own right, but rather an algorithm that we can run on any physical information processing system like a brain or computer. Set theory’s definition of 3 in terms of the empty set illustrates this recursive process. From this point of view, the ability to double x does not imply x’s existence, but rather shows that there are shortcuts to the process for some simple inputs (a one followed by many zeros). This type of shortcut is very common for all sorts of algorithms. The upshot of my view is that algorithms only exist when there’s an information processing system to execute them.

              Now to respond to some of your points…

              “The human race (and all intelligent life) will eventually die out, but to say that the highest number is just the highest one counted (or used) by intelligent life is perhaps to say that it is logically impossible for an intelligent creature to have counted higher in another possible world.”
              Personally, I don’t like the phrase “another possible world”. If there’s another world then we know nothing about it’s physics, and if we don’t know what rules it follows, then we have no way of speculating what may be possible in that world or whether it possibly exists. However, I will interpret “another possible world” to mean a situation that *seems* possible in our world. Okay, so the question is… what is actually possible in our world? Physics puts hard limits on what’s possible. When we say someone could have counted higher in another possible world, we’re actually saying “we can imagine someone counting higher in our world”, but the problem is our minds are good at imagining impossible things (Like a person who lives long enough to count to a Googolplex). I feel the logic jumps from “I am able to imagine it and follow various patterns” to “therefore it must exist independently of my thought processes”.

              Regarding the multiverse… In my view, if someone counts to M in another universe, that doesn’t affect whether M “exists” for us (because I don’t think numbers exist). It simply means this person executed their algorithm with a larger input. What *must* exist is that this other universe has enough information processing power to count to that number (i.e. to complete the algorithm). If there are P universes, I only think P has meaning as a number if some multiverse processing system runs the algorithm to count them. If no such creature or machine exists, then all that exists is the metaphysical multiverse, and P would have no *independent* existence.

              [About the platonic realm] “How do you know it might not just contain the numbers, but not things like a perfect horse, or a recipe that tastes “nothing like chicken”?”
              My point was that no one has *any* information about what it contains. However it does seem a little crazy to say that the platonic realm would only contain things that mathematicians are interested in! Which mathematicians would get to decide what’s interesting? I think this would arbitrarily put one’s own interests in a privileged position. Imagine a platonic realm containing only celebrity gossip. This is supported by the same evidence as one containing only mathematical objects. Should everyone be able to claim the existence of a platonic realm, filled with whatever interests them? If not, why would a mathematical platonic realm privileged over other fields of knowledge?

              “But just because we couldn’t access that information doesn’t mean it doesn’t exist – unless you define “exist” to involve interaction with our universe.”
              I agree, the definition of exist is crucial here. I do define exist to have some interaction with our universe. I think existence should be something which can be shown to have an independent existence to our imagination. Imagining a green rose doesn’t make it real, but if many people observe the green rose, repeatedly and consistently, and if we use scientific tools to verify that it’s green and a rose, then that should be considered to exist. “Existence” is usually a simple concept because there’s a sharp distinction between things that are constrained by physical laws (real objects), and things that aren’t (like fictional stories). If we were to imagine something which does not interact with our universe and say it “exists”, then it is not constrained by any known physical laws, and could be literally anything. Without strict constraints, “exist” doesn’t mean very much.

              In any case, if the platonic realm is real, then it *does* interact with the physical realm whenever someone does mathematics. Solving a mathematical problem is like peeking behind the curtain” You used the phrase “we are discovering stuff about numbers” through this platonic realm. This is only possible if the real world and platonic world could interact somehow. Enticing as this may seem, the laws of physics are causally closed, so we must wilfully deny some established truths in order to believe in this concept.

              “I know at least one philosopher who thinks that “reality” is maximally general – he has an interesting probabilistic argument for this.”
              Sounds interesting… I’m trying to google the definition of “maximally general” but I can’t find anything I understand. 🙂 What does this mean?

              “I don’t think the non-tangible aspect is specifically to explain a lack of evidence, but rather that that is what it seems like intuitively.”
              Yes, probably. I was being a little snarky with that comment. 🙂

              Another thing… is it possible that the platonic realm contains the assertion: 1=2? If not, why couldn’t it be there?

    • nsaranga

      Just some information that might be useful or interesting in the context of this post.

      While platonism about mathematical objects might seem ludicrous the reasons to believe
      in some variation of platonism is really found in the literature about mathematical explanations in scientific theories. It basically rests on some form or other of the indispensability argument put forward by Quine, about the ontological status of things used for explantion in our best scientific theories.

      Some interesting links for context:
      http://stanford.library.usyd.edu.au/entries/philosophy-mathematics/
      http://stanford.library.usyd.edu.au/entries/mathphil-indis/
      http://stanford.library.usyd.edu.au/entries/platonism-mathematics/

      This is not to say that nominalism about mathematical objects is indefensible or anything.

      • Reasonably Faithless

        Thanks for the links. I’ll have a look when I have some time…

        I actually don’t think platonism about mathematical objects seems ludicrous. In fact, mathematicians often act as though they are – when I’m sitting in my office trying to prove a theorem, it definitely feels as if I’m trying to learn about things that are out there somewhere, rather than just moving symbols around on a page according to a bunch of rules we’ve decided to make. (Obviously this says nothing about the truth of platonism.)

    • Gabe Czobel

      You have hit the nail right on the head! Craig and Sinclair are attempting to shift the burden of proof with respect to P1, and not only in the domain of mathematics. Regardless of how many different counterexamples they contend fail as defeaters of P1, P1 remains unsubstantiated. In order to substantiate P1 by this method, they would need to unequivocally eliminate all possible counterexamples while showing that they have exhausted all possible counterexamples, a pretty tall order, especially when dealing with notions like infinity.

      I find this typical of Craig’s style of argumentation in that he tries to shift attention away from his bald assertions by making much ado about how questionable counter assertions are, while making virtually no effort to actually substantiate his bald assertion. Although he doesn’t explicitly state that, consequently, his bald assertion stands, that is the implication he wants the reader to infer.

      If one comes to see these clever bits of misdirection in his arguments, it is easy to realise how anemic the arguments are.

      • Reasonably Faithless

        Haha absolutely! In “Infinite dreams”, I said:

        “Furthermore, the task of considering all special cases … must seem hopelessly impossible to Craig, since there are infinitely many special cases to consider. Even if Craig objected that the special cases … form a merely potential infinite collection, I’m sure he’d agree it had turned into an actual infinite once the task had been completed.”

        You’re right about the misdirection. It’s just amazing how pervasive it is, and how his followers don’t seem to notice and/or care.

    • Richard_Wein

      Hi. I’m in the “meaningless question” camp. What real distinction is being made when people ask whether numbers exist? Generally, none that I can see. I think this is a case of what Wittgenstein called language “idling”, i.e. not performing any real function. I suspect people feel that, because we can meaningfully question the existence of some things, it’s meaningful to question the existence of anything.

      I’m inclined to say that Craig’s P1 is meaningless too. But I don’t really care. His argument does no work at all. I have no more reason to accept P1 than to accept P3. P1 is a red herring. I wouldn’t waste my time trying to refute it.

      • nsaranga

        Hi Richard, I think most people find the “meaningless” answer a little uncomfortable because it doesn’t really answer how mathematical truths come to be true. This is especially the case with those people who think propositions are true because of reality. The hard question is what makes propositions that contain mathematical truths, true. It’s even more troubling when one of our most successful theories use highly mathematical descriptions of particles and spaces.

        • Richard_Wein

          Hi nsaranga,

          James didn’t ask the question “How do mathematical truths come to be true?”. So I don’t see my failure to answer that question as a deficiency! My point is that it’s a mistake to focus on the existence or non-existence of mathematical objects. If we want to address your question (“How do mathematical truths come to be true?”) we can do so without asking whether mathematical objects exist. My short answer to your question is that true mathematical statements are true just by virtue of the meanings of their terms. (Some would say that we shouldn’t describe mathematical results as “true”, but use some other word instead, such as “valid”. I think such a substitution can be helpful. But I don’t think it’s generally wrong to use the word “true”.)

          That said, I treated James’s question as “Do mathematical objects exist?”, when in fact he asked, “In what sense can it be said that mathematical objects like numbers exist?” I think the latter question is more meaningful, and I’ll give it a bit more of an answer. I would say that numbers (and other mathematical objects) exist–as concepts–in the sense that those concepts play a consistent role in our mathematical practice.

          • nsaranga

            Yep, I accept James didn’t ask that question, and that you didn’t answer it. It is just that the answer to the question of what makes mathematical truths true is pertinent to the the answer to the question of whether mathematical objects exist. If one accepts, what I think is closest to the folk view of truth, a kind of truthmaker or correspondence variant theory of truth, mathematical statements such as “2+2=4” must be made true by something. One could answer this your way by arguing that mathematics is just a highly precise language that has valid inferences(ie. true in all models, yet the word “truth” pops up again). I guess the problem with that answer is the role mathematical descriptions play in scientific explanations. How do we make sense of supposedly real things being described in purely mathematical language or have properties that are purely mathematical(ie. hilbert spaces, wave function, elementary particles) . Another problem is the, I think worrying feeling, of what exactly mathematical terms mean? It is not at all clear what mathematical terms mean, and whether they all have meaning. Meaning as intension and extension? Then of course there is the naturalist worry about mathematical objects as concepts existing as abstract objects? But perhaps that is not what you meant, but just that mathematical object are concepts that exists as electrical signals in the brain.

            • Richard_Wein

              “How do we make sense of supposedly real things being described in purely mathematical language…”

              It makes no fundamental difference whether empirical statements are in mathematical language or natural language. In fact there’s no clear dividing line between the two. Is the word “two” a word of mathematical language or natural language?

              Historically, people probably used numbers empirically (to count physical objects) before they started using them in more abstract ways, to make statements like “2+2=4”. It was the empirical practice which originally gave number words their meaning, so it’s not surprising that numbers can be used in modelling reality. Nor does the move from empirical use to more abstract use seem very surprising, when we consider the following sequence of sentences:

              1. One, two [counting sheep in one field], three, four [counting sheep in another field].
              2. I have two sheep in one field and two sheep in another field, or four sheep in all.
              3. Two sheep and two more sheep makes four sheep.
              4. Two and two makes four.

              People probably noticed that statements like #3 could be made for many different types of object (not just sheep), and so they eventually dropped the specific object and started making more general statements like #4.

              I find that thinking in these terms makes it fairly obvious what makes “2+2=4” true, much more so than trying to give a theoretical explanation.

            • nsaranga

              Hi Richard, one last post.
              Perhap my example of “2+2=4” was a bad example. What I was trying to get at was more complicated mathematics that we don’t use to count or are not applicable to anything outside a certain theory such as quantum theory. In fact

              But let’s go with “2+2=4”. It seems your inference from (3) to (4) is inductive(not in the mathematical sense). Inductive in the sense of, well we have got by counting things successfully in the past, I guess 2 and 2 makes 4
              all the time. If this is what you are saying, then it is a very strange way both getting to the meaning of mathematical terms and to getting to their truth. For starters I don’t know what form of sentence 4 is. There seem no objects to go with the properties. If you say the objects can be anything, then that makes it that we have discovered an a posteriori necessary truth, which certainly cannot be inferred by induction.

              If everything is the truthmaker for 2+2=4 then it is not clear what gives meaning to the term 2, 4 surely to make it 2+2=4 valid you must always put the put same object together with these properties. Hmm perhaps I’m a little confused at what you were trying to say.

            • Richard_Wein

              Hi nsaranga,

              I think “2+2=4” was a good example. It helps to look at a very simple case, where it’s easier to understand what’s going on.

              Let me clarify that my sequence (1)-(4) was a historical reconstruction, not an argument. My idea is that, if we understand why people came to use simple mathematical statements, we will better understand their nature. As well as thinking about the earliest uses of arithmetic, it’s also useful to think about how children learn to do arithmetic. I’ll just add that sentences (2)-(4) are a sequence of increasingly abstract or general sentences. (2) refers to a particular situation. (3) refers to sheep in general. (4) abstracts away any reference to specific things or types of things, giving us a sentence of pure mathematics, which can then be applied to many different real-world questions, by reintroducing specifics.

              Yes, my approach is an unusual one. I focus more on thinking about why and how people use language, rather than trying to give theoretical analyses of sentences. (Philosophers haven’t had much success with that!) If you want a name for my approach, I would call it broadly Wittgensteinian. (But I don’t claim Wittgenstein would endorse anything specific that I’ve said.)

            • Richard_Wein

              P.S. Returning to the earlier question, “Do numbers exist?”, the problem is that I don’t know how to understand the word “exist” in this context.

              The word has an established meaning within mathematical discourse, as when a mathematician might say, “There exists an integer between 2 and 4, namely 3”. It’s meaningful and true to say that the number 3 exists in that sense. But philosophers seem to want to step outside of mathematical discourse and use the word “exist” in some larger sense. What sense is that? The only other established sense I can think of is the one we apply to physical objects, such as when we say that horses exist but unicorns do not. Well, I don’t think that numbers exist in the same sense that horses exist, but neither do I think they fail to exist in the same sense that unicorns fail to exist. That sense of exist doesn’t seem to be applicable to numbers.

              So I’m left unable to answer the question, “Do numbers exist?”, because it doesn’t mean anything to me.

              The Platonist seems to be motivated mainly by the thought that sentences have to refer to objects, and therefore, since pure mathematical statements (like 2+2=4) don’t refer to any physical objects, they must refer to mathematical objects. But to me it seems unhelpful to talk of referring to objects, unless such talk has a purpose. When you talk about a man, say, it may be useful for me to ask, “Which man are you referring to?” But I have no motivation to ask what you’re referring to when you say “2+2=4”. The question would be meaningless.

              The basic problem lies in philosophers taking words out of their usual context, in which they are meaningful, and putting them into a new context where they no longer have any meaning.

    • Humberto Bortolossi

      Hi! Could you, please, inform the source of this? “But, according to a recent survey, (with 37.7% nominalists and 23.0% choosing some other option).”. Thanks!