One more on mathematical Platonism

IMG_8215Mathematical Platonism is one of those things I changed my mind about over time. And, of course, I may change it again. I began with curiosity and at least partial assent, convinced that mathematics is, indeed, uncannily weird. Too weird for there not to be some substance to the idea that mathematical truths are not invented by mathematicians, but rather “discovered,” in a way analogous to how astronomers discover planets (though, obviously, not literally: no telescope will ever show you the Pythagorean theorem…).

Here is an article I wrote for Philosophy Now back in 2011 where I sympathize with the Platonists. But by 2013 I strongly rejected the idea as presented by mathematician Max Tegmark (see this post at Rationally Speaking), and in 2015 I finally found what I still consider a good, non-Platonist, explanation of the weirdness of mathematics, as proposed by physicist Lee Smolin and summarized in this essay in Scientia Salon. Two more years have passed, so I guess it’s time to revisit the issue once more.

This time the springboard is provided by a critical paper by theoretical physicist Carlo Rovelli, entitled “Michelangelo’s stone: an argument against Platonism in mathematics,” published in the European Journal for Philosophy of Science (available for free here). Let’s take a look.

Rovelli asks whether there is a Platonic world M of mathematical facts, and, if so, what M may contain, precisely. In his paper he submits that if M is too large, it is uninteresting (in a sense to be explained in a minute), and if it is smaller and interesting, then it is not independent of us. Both alternatives — he says — challenge mathematical Platonism. He suggests that the universality of human mathematics may be a prejudice and illustrates contingent aspects of classical geometry, arithmetic and linear algebra, using them to make the case that what we call mathematics is, in fact, always contingent.

Rovelli begins by defining mathematical Platonism as “the view that mathematical reality exists by itself, independently from our own intellectual activities,” and acknowledges that it is a position accepted by some leading mathematicians, and so not to be taken lightly. He then introduces the idea of a mathematical world M of which he sets out to investigate the major characteristics.

First off, then, what does M contain? “Certainly M includes all the beautiful mathematical objects that mathematicians have studied so far. This is its primary purpose. It includes the real numbers, Lie groups, prime numbers and so on. It includes Cantor’s infinities, with the proof that they are ‘more’ than the integers, in the sense defined by Cantor, and both possible extensions of arithmetic: the one where there are infinities larger than the integers and smaller than the reals, and the one where there aren’t. It contains the games of game theory, and the topos of topos theory. It contains lots of stuff.”

But of course M — for a Platonist — also contains all the mathematical objects that mathematicians haven’t discovered yet. Rovelli says that M must contain the ensemble of all non-contradictory choices of axioms. “But something starts to be disturbing. The resulting M is big, extremely big: it contains too much junk. The large majority of coherent sets of axioms are totally irrelevant.” Ah, but irrelevant to what, and according to whom?

Here is where Rovelli introduces the metaphor that gives the title to his paper:

“During the Italian Renaissance, Michelangelo Buonarroti, one of the greatest artists of all times, said that a good sculptor does not create a statue: he simply ‘takes it out’ from the block of stone where the statue already lay hidden. A statue is already there, in its block of stone. The artists must simply expose it, carving away the redundant stone. The artist does not ‘create’ the statue: he ‘finds’ it.”

Sounds familiar? That’s pretty much what mathematical Platonists (or, really, Platonists of any stripe) claim. Rovelli acknowledges that, in a trivial sense, Buonarroti was right, since the statue is made of a subset of the grains composing the original block of stone. The problem is that there is an infinite number of such subsets, and that it is the artist that — arbitrarily — picks the subset that will then become the finished statue.

Here is an example of a Michelangelo statue, the horned Moses, that I photographed in the Church of St. Peter in Chain in Rome:

Mosè di Michelangelo, Basilica San Pietro in Vincoli

Rovelli goes on to say that one can tell the same story about the ensemble of all possible books, as recounted in the beautiful story by Jorge Luis Borges, The Library of Babel.

“Like Michelangelo’s stone, Borges’s library is void of any interest: it has no content, because the value is in the choice, not in the totality of the alternatives.”

Rovelli, intriguingly, goes further and says (correctly, I think) that science itself could be regarded in the same fashion, as the denotation of a subset of all of philosopher David Lewis’s possible worlds (see modal logic).

“What we call mathematics is an infinitesimal subset of the huge world
defined above: it is the tiny subset which is of interest for us. Mathematics is about studying the ‘interesting’ structures. So, the problem becomes: what does ‘interesting’ mean?”

The next step in Rovelli’s paper is to highlight the fact that interest, so to speak, is in the eye of the interested: “What is it that makes certain sets of axioms defining certain mathematical objects, and certain theorems, interesting? There are different possible answer to this question, but they all make explicit or implicit reference to features of ourselves, our mind, our contingent environment, or the physical structure our world happens to have.”

But, the Platonist will object, we actually expect any intelligent life form in the universe to arrive at the same mathematical truths that we have discovered. Well, do we? As a way to undermine this claim, Rovelli presents a number of examples in which what appears to be universal mathematics turns out, in fact, to be contingent on human peculiarities or on the human point of view. His case studies include: the geometry of a sphere, linear algebra, and the idea of natural numbers. All three are worked out in some detail, but I will pick just the latter for illustration.

“Natural numbers seem very natural indeed. There is evidence that the brain is pre-wired to be able to count, and do elementary arithmetic with small numbers. Why so? Because our world appears to be naturally organized in terms of things that can be counted. But is this a feature of reality at large, of any possible world, or is it just a special feature of this little corner of the universe we inhabit and perceive?”

The latter, argues Rovelli. First off, “the notion of individual ‘object’ is notoriously slippery, and objects need to have rare and peculiar properties in order to be countable. How many clouds are there in the sky? How many mountains in the Alps? How many coves along the coast of England? How many waves in a lake?How many clods in my garden? These are all ill-defined questions.”

Then Rovelli introduces a thought experiment that directly challenges the Platonist view:

“Imagine some form of intelligence evolved on Jupiter, or a planet similar to Jupiter. Jupiter is fluid, not solid. This does not prevent it from developing complex structures: fluids develop complex structures, as their chemical composition, state of motion, and so on, can change continuously from point to point and from time to time, and their dynamics is governed by rich nonlinear equations.”

The problem, however, is that there is nothing to count in a fluid environment, so our hypothetical Jovians will be highly unlikely to “discover” the natural numbers.

“The math needed by this fluid intelligence would presumably include some sort of geometry, real numbers, field theory, differential equations…, all this could develop using only geometry, without ever considering this funny operation which is enumerating individual things one by one. The notion of ‘one thing,’ or ‘one object,’ the notions themselves of unit and identity, are useful for us living in an environment where there happen to be stones, gazelles, trees, and friends that can be counted. The fluid intelligence diffused over the Jupiter-like planet, could have developed mathematics without ever thinking about natural numbers. These would not be of interest for her.”

So we may have a concept of integers because of the peculiarities of the Terran environment. Moreover, continues Rovelli, “modern physics is intriguingly ambiguous about countable entities. On the one hand, a major discovery of the XX century has been that at the elementary level nature is entirely described by field theory. Fields vary continuously in space and time. There is little to count, in the field picture of the world. On the other hand, quantum mechanics has injected a robust dose of discreteness in fundamental physics: because of quantum theory, fields have particle-like properties and particles are quintessentially countable objects.” So maybe the Jovian intelligence will arrive at the concept of integers, but only after having discovered quantum mechanics! As Rovelli puts it, “what moderns physics says about what is countable in the world has no bearing on the universality of mathematics: at most it points out which parts of M are interesting because they happen to describe this world.”

In the concluding part of his paper Rovelli writes: “Why has mathematics developed at first, and for such a long time, along two parallel lines: geometry and arithmetic? The answer begins to clarify: because these two branches of mathematics are of value for creatures like us, who instinctively count friends, enemies and sheep, and who need to measure, approximately, a nearly flat earth in a nearly flat region of physical space.”

Even the history of mathematics as a field of inquiry militates against Platonism: “the continuous re-foundations and the constant re-organization of the global structure of mathematics testify to its non-systematic and non-universal global structure. … Far from being stable and universal, our mathematics is a fluttering butterfly, which follows the fancies of us, inconstant creatures. Its theorems are solid, of course; but selecting what represents an interesting theorem is a highly subjective matter.” Just like picking an interesting book from Borges’s library, or carving a statue the way Michelangelo did.

“It is the carving out, the selection, out of a dull and undifferentiated M, of a subset which is useful to us, interesting for us, beautiful and simple in our eyes, it is, in other words, something strictly related to what we are, that makes up what we call mathematics.”

So here is where I stand now on the matter: the joints efforts of Smolin and Rovelli amount to a hook and an uppercut punch to mathematical Platonism. It isn’t a knockout blow, but it’s pretty close. Then again, I changed my mind on this in the past. It may happen again.

277 thoughts on “One more on mathematical Platonism

  1. saphsin

    “we wouldn’t be able to have philosophical discourse to convince others in the first place.”

    sorry to clarify, meaning it would take something additionally on a empirical basis to convince others, if it wasn’t a matter of self-contradiction

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  2. Daniel Kaufman

    Coel:

    Philosophical positions are not the sort of thing for which there is nearly ever widespread consensus or overwhelming reasons/evidence. Philosophy is just not that kind of animal. If you actually were open to learning from others, you’d have learned that by now, not just from me, but from Massimo and others.

    As for expertise, you can think whatever you like. The trouble is that what you think doesn’t matter. Without the relevant education, degrees, and peer-reviewed publications, no one on the earth will pay you one penny to teach philosophy, and no one who matters will care what your views are on philosophical subjects.

    Like it or not, this is the consequence of the professionalization and disciplinization of scholarly inquiry. So it doesn’t matter what you “defer to.” You won’t be the one setting the agenda in the teaching of ethics, aesthetics, metaphysics, epistemology, etc., to generations upon generations and thousands upon thousands of university students.

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  3. synred

    possible is the word! Possiblities can exist w/o being actualized.

    I doubt the ”best possible surgical’ technique exist. Surgery is too complicated for that .. different cases maybe better suited to techniques. The possibilities are endless, there is no best. What docs seek is good enough.
    t
    th

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  4. Disagreeable Me (@Disagreeable_I)

    Hi Dan,

    is one of those “not even wrong” ideas.

    Or more charitably, your interpretation of what I am saying is ‘not even wrong’. You might instead want to ask me how I make sense of the interaction between the mind and the physical world rather than assuming I am so badly confused.

    given the interaction of our conscious thoughts and our bodily activities, minds could not be abstract objects, even if they were objects.

    There is no interaction between physical and abstract objects. That’s not the relationship. Physical objects instantiate abstract objects. A coin does not have to interact with the concept of an abstract circle in order to be circular. It just instantiates the pattern. The brain instantiates the pattern of the mind, and it is the action of the brain that causes limbs to move and so on. But I would locate my consciousness with the pattern rather than with the physical brain, such that I would be open to the concept of mind-uploading in principle, for example.

    I am quite sure that if I followed you around for several days or so, I’d find that you act and talk like everyone else does

    Indeed you would.

    and not as one would if he actually believed the bizarre things you claim to be committed to

    How do you think such a person would behave, if not like an ordinary person?

    As I have been saying, our ordinary concepts of existence and so on serve us perfectly well in the everyday world. No situation has ever arisen in my life where my views would make any differene, save in philosophical debates. About the only practical consequence so far is that I no longer wonder why the world exists and I am no longer puzzled by philosophy of mind.

    However the difference would make itself known in certain scenarios which are currently science-fiction. If I were to encounter an intelligent computer which processed information in a manner analogous to a human brain, I would treat it as a conscious being. You would not. I might consent to Star Trek teleportation or mind-uploading. You would not.

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  5. Daniel Kaufman

    Possible worlds are not like foreign countries. They are just a literary device for speaking of what could have been the case, but isn’t. To speak, then, of possibilities as “existing” is to commit a basic error. Beyond which, for the reason that Quine indicates, it’s not clear that one even can identify principles of individuation for “existing possibles,” which makes nonsense out of any talk of them.

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  6. Daniel Kaufman

    However the difference would make itself known in certain scenarios which are currently science-fiction. If I were to encounter an intelligent computer which processed information in a manner analogous to a human brain, I would treat it as a conscious being. You would not. I might consent to Star Trek teleportation or mind-uploading.

    = = =

    That you are fixated on such examples is the problem. Just as hard cases make bad law, this sort of speculation makes for very bad philosophy.

    Liked by 1 person

  7. synred

    why is this so confusing to everybody?

    There are lots of possibilities, most of which will never be realized in practice. Sets of axioms have theorems that could be derived from them, mostly they won’t be because nobody thought of these axioms.

    I don’t need Plateau to understand that.

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  8. couvent2104

    DM

    In order for me to accept that a unicorn exists, you have to define it robustly like a mathematical object.

    Ah, but not all mathematical objects are robustly defined. Pascal(°) already noticed that you have to start from somewhere, i.e. from undefined terms. Unicorns play the role of an undefined term in my little axioms system, just like “point” and “line” are undefined terms in a typical axiom system for Euclidean geometry(°°). If you think that mathematical points and lines “exist” there’s little reason to deny the “existence” of unicorns.

    Of course, once you started from somewhere, you can give robust definitions of things like triangles etc. in Euclidean geometry. But they are based on undefined terms.

    (°) It’s 30 years since I read him, so forgive me if I’m wrong and it wasn’t Pascal.
    (°°) As Bunsen pointed out, there are several axiom systems for Euclidean geometry, but each time you have to accept undefined terms.

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  9. Daniel Kaufman

    There is no interaction between physical and abstract objects. That’s not the relationship. Physical objects instantiate abstract objects. A coin does not have to interact with the concept of an abstract circle in order to be circular. It just instantiates the pattern. The brain instantiates the pattern of the mind, and it is the action of the brain that causes limbs to move and so on. But I would locate my consciousness with the pattern rather than with the physical brain, such that I would be open to the concept of mind-uploading in principle, for example.

    = = =

    This is so breathtakingly ignorant, takes so many horrifically difficult things for granted, and involves so many outright howlers that it would take an entire semester’s Independent Study to disentangle it all. But I’m glad you said it, as it absolutely confirms to me that I’ve been right in my perception of your philosophical acumen. If you really have any interest in developing serious views on this subject, you desperately need to be reading and listening much more and talking much less. You are not yet at the level of understanding in these areas to say anything that is really worth discussing, beyond a few minutes’ casual chat.

    I strongly encourage you to spend some time with the foundational literature before branching off into specialty areas like cognitive science. This is best done through introductory level courses. My own Introduction to Philosophy course, which I’ve been teaching now for over twenty years, can be watched in video form entirely for free on Itunes U. It wouldn’t be a bad place to start.

    https://itunes.apple.com/us/course/introduction-to-philosophy/id655626248

    But the sort of stuff you’re trading in, now? Honestly, it’s really not worth your time or anyone else’s. I know that’s tough to hear, but it’s better than being lied to and disingenuously flattered.

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  10. Disagreeable Me (@Disagreeable_I)

    Hi Massimo,

    No, unicorns definitely do not exist, and to raise the possibility is to fundamentally equivocate on what “exists” means. Let’s not pull any Clintons (Bill) around here.

    I didn’t raise the possibility. Couvent did. I was only answering him.

    So, I would say if anyone is equivocating, it would be Couvent (not that I personally mind or would fault him for it), who did not specify what sense of ‘exist’ he meant. I’m the one who is insisting that the interpretation of the concept of existence is tricky. I’m the one who is being clear about what sense of existence is intended and explaining in depth in what sense(s) unicorns might be said to exist.

    Do unicorns exist in the every day sense, i.e. physically, on planet earth? Of course not.

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  11. Bunsen Burner

    Coel:

    I have to admit to having great difficulty parsing your views on mathematics. It’s not obvious to me how to interpret what you write. However, I am willing to take one last stab at what i think will be a knock down argument based on exactly what you said would convince you; namely:

    ‘What would contradict me is widespread use of axiom systems that are blatantly inconsistent with real-world behaviour.’

    Interpreted reasonably of course. Mathematicians have known for a long time that an axiom system isn’t enough for doing maths, you need a model too. A result known as the Löwenheim–Skolem theorem however proves that the number of models of any first order axiomatisation is infinite. For example, Peano’s axiomatisation of arithmetic allows for a countably infinite number of so-called non-standard models of arithmetic. This of course is complete esoterica to most people. But these different arithmetics exist and are different to standard arithmetic, they are studied, and some mathematicians even debate which arithmetic model is “correct”.

    So… in a precise model-theoretic sense there does exists widespread use of axioms/models which are inconsistent with the real world. An infinite number in fact whereas the real world counterparts are going to be finite.

    Check and mate I believe.

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  12. Disagreeable Me (@Disagreeable_I)

    Hi Dan,

    Possible worlds are not like foreign countries. They are just a literary device for speaking of what could have been the case, but isn’t.

    I understand that is how they are usually interpreted. This is not a misunderstanding or confusion on my part. Yet I claim that all possible worlds are actual. I am a modal realist.

    This is so breathtakingly ignorant

    I’ll defer to you that this is a possibility, yet your comment is not terrifically helpful, because it doesn’t do anything to help me see where I might be making mistakes. I agree that it would be good for me to read more of the literature, and in an ideal world I would. I’m trying to educate myself as much as time permits, but it’s slow going. I’ll instead interact with those who give me more specific feedback, if you don’t mind.

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  13. Disagreeable Me (@Disagreeable_I)

    Hi Couvent,

    Ah, but not all mathematical objects are robustly defined. Pascal(°) already noticed that you have to start from somewhere, i.e. from undefined terms.

    Yes, that’s fine, I know. But your unicorn axioms just amount to “there exists an x”. You’d have to add thousands or millions more to have anything like a recognisable model of a unicorn. As I said, you’d have to have what amounts to a computer program to simulate unicorns and their environment. If we ran such a program, we might recognise the output as depicting unicorns. My view is that the unicorns in the simulation are abstract mathematical objects and they exist, and they are unicorns to the extent that humans would agree that they resemble unicorns. Perhaps only superficially if they just have the outward shape of unicorns, but more profoundly if they turned out to be molecule-by-molecule simulations of animals resembling horned horses.

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  14. Coel

    Dan,

    Philosophical positions are not the sort of thing for which there is nearly ever widespread consensus or overwhelming reasons/evidence. Philosophy is just not that kind of animal. If you actually were open to learning from others, you’d have learned that by now, not just from me, but from Massimo and others.

    I am aware of that exposition of how philosophy is. And I am even happy to grant it as correct. But the consequence is then that — as I see it — one can legitimately doubt any claims being made.

    For comparison, I accept the claims and expertise of aircraft engineers over those of, say, magic-carpet enthusiasts, because the former can produce overwhelming evidence of flying aircraft (whereas evidence of actually-flying magic carpets is meagre).

    Now, if the aircraft engineers could not produce abundant evidence of properly functioning aircraft, and instead spent all their time arguing over whether a particular aircraft would or would not fly, while not providing evidence one way or the other and never settling the issue, then I would legitimately doubt whether they had any actual expertise in making aircraft that fly. Essentially, I’d have no basis for granting the claim.

    By the way, the “we get paid to teach it” rebuttal doesn’t impress either, given that the same could be said by cabals such as theologians, post-modernists, and various loony ideologies that are all the rage, including various versions of feminist ideology, race theory, critical theory, blank-slate sociology, and other SJW fads.

    Liked by 1 person

  15. Daniel Kaufman

    Well, given that a reply to DM that I spent some time on was deleted, I will bow out. It was substantive, responsive, tough, but fair.) As always, if anyone wants to communicate with me privately on these subjects, feel free to email me at Missouri State. Otherwise, the field is left, once again, to Coel and DM.

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  16. Disagreeable Me (@Disagreeable_I)

    Hey Dan,

    I know it’s annoying when a comment you’ve worked on is deleted or lost, so if it’s any consolation I read it before it was deleted.

    I take it in the spirit in which I assume it was intended: friendly advice, so thanks for that.

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  17. brodix

    DM,

    Dimensionless points are mathematical objects. They are also mathematically self negating, being a multiple of zero.
    The same applies to lines and planes, with a zero dimension. The same applies to numbers as well. They are abstractions of what are supposed to be universal concepts. What would be a universal 1? Other than a monotheistic deity, or the Big Bang universe? Actually, now it would just be a monotheistic deity, because we have multiverse theory! The one has become a multiple of one. As soon as you have a singular unit, then there are other such units and any particular unit is no longer universal.

    How about two as a universal object? Is the one on the left the one and the one on the right two? Or other way around? Once we really start examining these ideas and not just take them for granted, they start getting fuzzy.

    Then for the idea of chess as timeless. Chess is a process. Start, lots of moves, finish. How is that timeless? As I pointed out previously, math likes to think of itself as timeless, but operations are verbs.

    The fact is that our minds function by developing concepts, that we try to fit our experiences into. Like the idea of a point, or a line, or a game. To assume these concepts are fundamental to reality, rather than mental constructs to function in reality, would seem to be a conceit, more than a logical argument. They are maps. The territory is dynamic.

    As for expertise, it’s probably safe to say that governments are full of experts, from politicians and bureaucrats, to all the various people working for them, yet they periodically have these rituals, called elections, that they seem to take quite seriously, where they actually go out and ask the general public how they are doing and which groups of other such professionals they would prefer.

    We also have a legal system, that is also quite full of people who have devoted their lives to the study of the law, yet at its core is this process called a trial by jury, where they just go out and get twelve dolts to make the serious decisions. How completely idiotic is that?

    Obviously the general public are not experts on the issues in question, so why would they do this? Any ideas from the expert philosophers, since this would seem to be the sort of topic philosophy should be knowledgable about. No?

    From my totally uneducated point of view, it might be that expertise naturally creates a form of bias, as people spend their entire lives focused on a particular field, or sub-set of a field, or even sub-set of a sub-set of a field.

    So unless one is building airplanes, where the result is evident, having a way to step back and find some method of putting it in a larger context seems necessary.

    The alternative to the expert is the generalist and possibly it is not complete coincidence those who run armies are called generals, while specialist is an enlisted rank.

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  18. Robin Herbert

    The central ‘marble’ metaphor reminded me of a discussion I had some time ago concerning a particular variant of the Monty Hall problem in which the success of the strategies depended upon the strategy of the host.

    To illustrate a point I wrote a program to calculate the probabilities of winning against the cost of prizes per game for the host and then opened the results in a spreadsheet to do a scatter plot.

    I pretty banal exercise, I am sure you would agree, but here is the scatter plot:

    https://photos.google.com/share/AF1QipMPlzpY4ne7tn38rR1iPtcoeix2SobtZ1-c-ZIdcDFvTrzDykXX1irZQ-zKZR4mBQ/photo/AF1QipPOnmz6kXnNKV7oMQAwaJ_BrJ2wPys0bY5Np1Vu?key=OEJ3bmJyenNkWDlzSlRJWHpIckZlN0Q5bDRmbjJ3

    Now it would be fair to say that I discovered that pattern – it was about the last thing I was expecting, a rather beautiful pattern emerging from some rather prosaic data.

    People sometimes say, with wild poetic license, that the sculptor ‘discovers’ the sculpture in the marble. But I didn’t discover it in that sense.

    I did not set out to draw that particular pattern or anything like it.

    Neither did Michelangelo give a couple of desultory knocks at a piece of marble and say ‘Blow me! A horned Moses!’

    No, I really did discover that pattern. So the metaphor of the marble is not apt. For similar reasons, neither is the Library of Babel.

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  19. Robin Herbert

    Hi synred

    “why is this so confusing to everybody?”

    People disagreeing are not necessarily confused.

    But if this is so simple can you answer my questions about the example I gave earlier of two sets of chess players independently coming up with a new Queen rule added to standard chess.

    Did the mathematical facts about the new rule become true just once when the first game was played?

    Or did they become true twice that day, once for the first game and once for the second?

    Or were those facts true in any case?

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  20. Robin Herbert

    …blank-slate sociology, and other SJW fads

    The “blank-slate sociology” that, for example, Pinker criticises is actually a claim from sociobiology, so yeah agree with you there.

    I don’t think Pinker realised it was a claim from sociobiology. He seemed to think that blank state sociology was a denial of the influence of evolution. There I go, disrespecting my elders and betters.

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  21. Robin Herbert

    I do think that it is interesting that there might be a physicist and aircraft engineer who is a senior academic at one of the worlds most prestigious universities who runs a series of experiments to which the prestigious university has dedicated an entire department to for nearly three decades and who finally publishes a paper showing evidence in favour of the hypothesis with overwhelming statistical significance, but I can read just a few lines of the intro on the departments web page and know for certain that the entire thing is complete nonsense.

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  22. Robin Herbert

    I will try to semi formalise Rovelli’s notion of uninteresting to show why the set of axioms is not uninteresting in this case.

    The set of all strings of length n is uninteresting because I know that changing one bit will have the predictable result of one bit having been changed. This is the block of marble or the Library of Babel.

    The only thing to be discovered in the changing of one bit in this set is that one bit has been changed. In other words there is nothing to be discovered.

    The set of all strings of length n treated as axiom sets will be not be uninteresting in this sense because the changing of one bit might have radical and unexpected results which could not have been predicted by the action itself.

    So there is something to be discovered in that you can discover things about them that you did not know you would discover.

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  23. Robin Herbert

    From the paper:

    Shouldn’t we expect – as often claimed – any other intelligent entity of the cosmos to come out with the same “interesting” mathematics as us.

    No, or at least why should we expect this? Who has claimed this?

    It has often been claimed that a non-human intelligence could arrive at the same axioms and would arrive at logically equivalent conclusions. This is usually in response to the claim that mathematics is a purely human activity.

    But I have never heard anyone claim that any other intelligent entity would come out with the same mathematics as us. Perhaps someone can cite this claim being made.

    So Rovelli is spending the next three or so pages demolishing a straw man. Even then he appears to be saying that anyone who started with a spherical geometry rather than a rectilinear one would be using a different mathematics. Not so, they would be using the same mathematics, using a different basis.

    And the Ancient Greeks, despite what he seems to think, were well aware that the results of plane geometry were dependent on rectilinear assumptions, as can be seen in Aristotle’s physics.

    In any case it is all moot, there is nothing inconsistent to Platonism in other beings becoming interested in different types of mathematics depending on their intelligence, psychology, physiology, environment etc.

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  24. synred

    Burner:

    >Interpreted reasonably of course. Mathematicians have known for a long time that an axiom system isn’t enough for doing maths, you need a model too

    https://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic

    Well at least I learned something new from this seemingly overheated discussion.

    I don’t understand what ‘model’ means in the context. They seem to had additional axioms, so I’m not sure how that is relevant. Regular arithmetic seems to be a sort of sub-set of the infinite number of possible ‘models’. That these ‘models’ have any use is not clear. I’ve been able to do math w/o even knowing about them.

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  25. Robin Herbert

    Note also that Smolin’s mathematical truths appear to be geographical as well as temporal.

    A mathematical fact that has already been true in Constantinople for some time may not yet be true in London.

    That seems to me to be buying a desert ontology a slum epistemology.

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  26. Bunsen Burner

    synred:

    ‘Well at least I learned something new from this seemingly overheated discussion.’

    Glad you liked it. I had to call in a favour from a Phd to put it together, at it strains my knowledge of the subject matter. I just hope Coel appreciates the effort 🙂

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  27. Disagreeable Me (@Disagreeable_I)

    Hi Brodix,

    You addressed this to me but it doesn’t seem to have anything particularly to do with anything I said. I’ll answer a couple of points anyway.

    They are also mathematically self negating, being a multiple of zero.

    This to me seems to be nonsense. I don’t accept that dimensionless points are “self-negating”, whatever that means. Nor are they a multiple of anything. A multiple is one number multiplied by another number. A dimensionless point is not a number multiplied by another number. And anyway, zero is a perfectly valid mathematical concept anyway, and zero is not “self-negating” as far as I can see.

    Then for the idea of chess as timeless. Chess is a process. Start, lots of moves, finish. How is that timeless?

    First, the word ‘game’ is ambiguous, and can refer to either a ruleset or a playthrough, so I’ll use the latter terms.

    The ruleset can be regarded as timeless because it doesn’t change. Or if it does, I would say you have just discovered a slightly different timeless ruleset.

    Any individual playthrough of chess is a process from one perspective, but from another perspective it’s just one of the possible playthroughs, each of which can exist timelessly. When you’re playing a game, you’re going through the motions and tracing out one of these pre-existing playthroughs. The same way that derviing Pythagoras’s theorem is a process from one perspective, but once it’s written down it can be regarded as a mathematical object that exists timelessly.

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