You are being too modest to call it "just" philosophy of math. It's important to get the background right and I am relatively naive in this area, so I appreciate your attempts to pin me down.

I am disturbed by your suggestion that I am a platonist. To me, platonism suggests the existence of an abstract world, independent of the physical world, to which we somehow have access through mathematical intuition. I definitely want to deny that. I would be happy to be called a mathematical realist, in that I think there is a fact of the matter about what should be called a mathematical truth, but I want to cash that out in terms of the physical world rather than some abstract mathematical world. In any case, I think it is helpful to distinguish platonism from the broader concept of mathematical realism.

As I said towards the beginning of the essay, I have a somewhat pragmatic notion of truth, at least in the context of science and math, so I am prepared to accept something as "true" if it makes the system of knowledge hang together more efficiently than it would do otherwise. This notion is partly empirical, in the sense that empirical facts form part of our knowledge network and so our axiom systems have to be chosen to make those facts fit efficiently into the network, but it is also partly pragmatic, i.e. there are several choices for how to build theories that connect up our knowledge, but the "true" one is the most useful, by which I mean the most efficient. Because of this, even though the notion of a theorem is not itself naturalistic, the set of things we should be willing to call the true theorems of our world is.

Regarding claims that are not theorems, if they are true claims then they are theorems of a meta-theory. I would argue that the axioms chosen for the meta-theory are not arbitrary, but decided by the same sorts of considerations as the axioms of the lower level theories. I realize that there is an infinite regress here, but where to stop bothering about deciding truth claims in higher level theories is again, I would say, a pragmatic matter. For example, I want to say that there is a fact of the matter about whether we should accept the axiom of choice, because without it we can't do conventional analysis and that would be a disaster for many areas of mathematics and physics. For things that occur at a much higher level of abstraction there are two possibilities: either they do have a truth value that is not yet determined because we don't know how they impact out other theories yet, or they don't because they will never be hooked up to the rest of knowledge in any significant way. For any given statement, we don't know which category it is in, and I am prepared to say that there may be some statements that do not have well-defined truth values, but we can't know which ones those are.

By the way, I have been implicitly assuming throughout this that there is a unique "most efficient" encoding of our knowledge in a scale-free network, which is what we are trying to generate with our theorizing. I don't think it would matter too much if it was not quite unique, but could be modified without changing the overall structure too much. However, it is possible that there are two or more very different ways of generating an efficient knowledge graph that incorporates all of our empirical knowledge. If so, then truth claims would be relative to that. However, since under-determination rarely occurs as a practical problem, I doubt that this is the case in our world.

"Since mathematical theories are derived from the natural world..."

No. Just like physical theories, they are derived from initial assumptions (axioms, postulates) that could be arbitrary and false. In Big Brother's world, a new arithmetic theory has been derived from Big Brother's postulate "2+2=5":

"In the end the Party would announce that two and two made five, and you would have to believe it. It was inevitable that they should make that claim sooner or later: the logic of their position demanded it. Not merely the validity of experience, but the very existence of external reality, was tacitly denied by their philosophy. The heresy of heresies was common sense. And what was terrifying was not that they would kill you for thinking otherwise, but that they might be right. For, after all, how do we know that two and two make four? Or that the force of gravity works? Or that the past is unchangeable? If both the past and the external world exist only in the mind, and if the mind itself is controllable what then?"

There are paradoxes in the new arithmetic theory. Here is one of them (it can be juxtaposed with the twin paradox in Einstein's theory of relativity):

3(2 + 2) = 3x5 = 15

3(2 + 2) = 3x2 + 3x2 = 6 + 6 = 12

Pentcho Valev

    Your essay view of math follows mine nearly exactly.

    Mine is a little stronger on the in the natural world because it is apart of the natural world like gravity. I also add the idea of fractals (self similarity) rather than hubs. Thus math developed out of our human scale and applies to other scales because the universe is fractal.

    There are some problems with the abstractions that deviate with the observation of math such as irrational numbers, division, and infinity. These things are not observed in our scale and are part of the human introduced postulates that are false. Indeed, the current study of math allows the introduction of postulates and the reasoning from those postulates. It is called ``pure'' but it is really only unjustified abstraction. This is not necessarily physics. For example, the introduction of non-Euclidean geometry is unreal - its use in cosmology is problematical because the universe has been measured to be flat (Euclidean).

    My view allows the idea of using a mathematical structure that is observed such as by statistics or group theory to be considered real. For example, the periodic table was developed first by noting common characteristics of elements. A few holes were filled (predicted) by where the hole was in the classification scheme. Later, the causal underlying structure of atoms explained the periodic table. Indeed, the position of an element indicated something about the atomic structure. The same type of classification is true for the particle group models. Holes in the group model have been used to predict particles that were found. Can this be used to imply an underlying structure of particles? My model says yes.

    Matt,

    Though I couldn't disagree with you more, I really enjoyed your essay.

    We will have much to debate -- my view agrees with Max's, and my own upcoming essay deals with the issues of Godel and Goldbach that Tim raised.

    Two things for the time being:

    1. "There is no 'adding zeroes and ones to the end of binary strings' research group in any mathematics department. " Sure there is. Chaitin's number is maximally unknowable, and its algorithm cannot predict the next binary digit of the value. What's more, the value is dependent on the language in which the algorithm is written.

    2. Hierarchical knowledge? What if knowledge is laterally distributed on multiple scales in the hub-connected complex network? No hierarchy -- which was Bar-Yam's solution to the problem of bounded rationality (Herbert Simon).

    Tom

      Dear Matthew Leifer,

      At certain times I take a stance similar to this. You might by way of comparison look at Peterson's paper for a different perspective. In my paper I am primarily concerned with what I call mathematics that has "meat" or "body," by which I mean things that are computed in some rather explicit way and that have reference to physical properties. I look informally at decidability issues, by treating this in a somewhat physical way, and make arguments with respect to the complexity of numbers.

      There is what might be called the "soul" of mathematics, which is all of that Platonist stuff. I am not committed to saying this exists or does not exist. This is in some way connected to mind or consciousness, but connected in way that I don't understand and I don't think anyone else does either. Whether one want to argue for the existence of this "soul" is a matter of choice or almost what might be called faith. I don't think there ever will be some decidable criterion whereby we can say Platonia exists or not. I will put on the hat of Platonism at times and at other times not wear it. In my essay I largely keep it off.

      "On a dark night in a city that knows how to keep its secrets; on the tenth floor of the atlas building one man searches for answers to life's persistent questions, Guy Noir private eye." Garrison Keillor "Prairie Home Companion. That about states where the deep question about the relationship between mathematics and physics lies.

      LC

      I won't get into deeper waters here, but just a note: I think you are being too scared of the idea of "mathematical intuition". An intuitive judgment is just one that you accept without further argument, so all of our reasoning ultimately relies on accepting intuitive judgments. This includes logical inference: Given "A" and "If A then B", you are happy to accept "B" by logical intuition. If you think you have to back up accepting the conclusion by further argument, you get into the situation in "Achilles and the Tortoise". So accepting non-empirical mathematical intuition is no more (or less) scary than accepting non-empircal logical intuition, which you are committed to. One might describe this as saying you believe in non-empirical logical facts that your mind can intuit. I see no problem with that at all. But if that's right, why a problem about non-empirical mathematical facts your mind can intuit as well? Of course, to do this well you have to be thinking in terms of sharp mathematical concepts, just as you have to understand the logical concepts. It is just that drawing a line between math and logic, with one obviously OK and the other somehow problematic, seems unmotivated.

      1. I was referring to the specific (very boring) formal system that I had introduced earlier. There is no mathematical research group studying that. I initially thought to include a footnote that of course there are people studying the general structure of formal systems, which would cover the kind of thing you are talking about here.

      2. I am not sure I understand all of the technical terms you are using here, but indeed I do expect that knowledge is distributed on multiple scales. I am not familiar with the literature on bounded rationality, but it sounds like something I should look into.

      If you don't think physical theories are derived from the natural world, then I don't think we have much to talk about. Of course, other factors go into the construction of our physical theories, but if they are not designed to account for the empirical facts then what is the point of them?

      4 days later

      Hello. This is the second essay I read after that of Lee Smolin, that tries to give a naturalistic philosophy of mathematics, as opposed to a Platonistic one. As I commented there, I have yet to see a coherent formulation of naturalism. The Stanford encyclopedia article you refer to on this point, admits it directly : "The term 'naturalism' has no very precise meaning in contemporary philosophy." It is remarkable to see that still after quite a long time that a majority of physicists and philosophers who care about metaphysics are trying to defend and develop naturalism, no clear formulation of its actual meaning could even reach a status of notability without being also loaded with big troubles (such as those of Bohmian mechanics); while I would explain the lack of well-known coherent formulation of idealism by the lack of serious tries by competent physicists and philosophers, a gap I care to fill by my essay.

      The main difference I see between yours and Lee's exposition, is that Lee entered in specific details, especially about where he draws the line of existence between things. His claims are totally incoherent, both logically and with respect to existing knowledge in math and physics as I explained in my many comments there, however he has at least this merit, of taking that risk of making specific claims and thus expose himself to refutation. It seems you found a pretty good way to minimize the amount of criticism that you will get, by minimizing the quantity of claims contained in your essay. Indeed, most of the ideas there are already given in the abstract. You managed to approach the standard size of 9 pages not by adding up many ideas but rather by diluting the few you had. Anyway, by lack of effective matter to discuss from your essay, and also because I don't like to repeat myself, I invite you to read my comments to Lee's article (I'll still have more to write there), for the general remarks I made about this common topic between yours and his essay, of naturalism vs. Platonism, and for the specific analysis of his tries to effectively specify the claims of naturalism (I'm curious how would your options differ from his).

      Now for your few specific claims :

      You wrote: "Naturalism is the position that everything arises from natural properties and causes, i.e. supernatural or spiritual explanations are excluded. In particular, natural science is our best guide to what exists, so natural science should guide our theorizing about the nature of mathematical objects."

      It is true that sciences of "natural" things (physics, biology) were extremely successful and these aspects of the world are a privileged field for scientific inquiry. However, like so many naturalist authors, you fail to distinguish between the scientific method and materialistic metaphysical prejudices, by using the ambiguous expression of "natural science" that may actually mean either the scientific method or the body of discovered knowledge in physics and biology, but where the adjective "natural" it contains can be irrationally played on in philosophical discussions to make it look as if the scientific method required to adopt such metaphysical prejudices. It doesn't.

      "A naturalistic theory has no place for a dualistic mind that is independent of the structure of our brains. Therefore, if we have intuitive access to an abstract realm, our physical brains must interact with it in some way. Our best scientific theories contain no such interaction."

      Our best scientific theories do not contain any explanation of anything psychological : what it may mean for a mind to understand something in general. Thus, there is no wonder why they do not explain either how we (I mean, some people) can understand mathematics in particular.

      "The only external reality that our brains interact with is physical reality, via our sensory experience. Therefore, unless the platonist can give us an account of where the abstract mathematical realm actually is in physical reality, and how our brains interact with it, platonism falls afoul of naturalism."

      The content of this argument looks like an absolute argument against platonism, to which I replied in comment to Lee's article. Strangely, for no reason I could see, the last phrase "platonism falls afoul of naturalism" looks as if it was only an argument relative to naturalism (thus void outside it). But then in this case, it implicitly seems to assume that a remark that "Naturalism contradicts X" should be taken as an argument against X, thus taking for granted that naturalism should be true. However I still beg for a rational argument for naturalism, in the name of which it would make sense to consider a view as being made less plausible just by its conflict with naturalism.

      "The various attempts to reduce all of mathematics to logic or arithmetic reflect a desire [to] view mathematical knowledge as hanging hierarchically from a common foundation. However, the fact that mathematics now has multiple competing foundations, in terms of logic, set theory or category theory, indicates that something is wrong with this view." ; and in your comments: "What I mean is rather that which formal systems we decide to call mathematics, out of all the myriad of arbitrary axiom systems we might choose to lay down, is a matter of physics."

      Well no. There is, for example, a clear absolute sense of which rules can form a valid and complete system of proofs for first-order logic, and satisfy the completeness theorem. It is clearly independent of physics. Now for the choice of an axiom system to best serve as the foundation of mathematics, I do not see such a competition between possibilities as you assume. Instead, as I explained in my work on the foundations of maths which I developed showing its intrinsic necessities independently of physics, I rather see a coherent whole of complementary parts of the foundations, with a dynamic articulation between them. Where there are alternative possibilities in competition for the same purpose (such as axiomatizations of set theory) I see a situation where in some aspects there is a preferred kind of formalization, and in other aspects such as the continuum hypothesis there is a remaining real diversity of acceptable systems corresponding to a real diversity of acceptable realities, that can often be understood as due to the real fact of ambiguity of the powerset of infinite sets.

      "Firstly, in network language, the concept of a "theory of everything" corresponds to a network with one enormous hub, from which all other human knowledge hangs via links that mean "can be derived from". This represents a hierarchical view of knowledge, which seems unlikely to be true if the structure of human knowledge is generated by a social process. It is not impossible for a scale-free network to have a hierarchical structure like a branching tree, but it seems unlikely that the process of knowledge growth would lead uniquely to such a structure. It seems more likely that we will always have several competing large hubs..."

      If things went as you describe here, we would not have got the amazing success of mathematics in physics we had, with an amazingly universal agreement on what is the right fundamental theory of gravitation on the one hand, of microscopic physics on the other hand, how amazingly well, each on their side, they indirectly explain so many things on almost all physical processes that could be tested. So I do not see your philosophy coherent with the state of science that could be observed.

      "...and that some aspects of human experience, such as consciousness and why we experience a unique present moment of time, will be forever outside the scope of physics." What a nicely non-naturalistic claim ! ;-)

      "I have argued that viewing mathematics as a natural science is the only reasonable way of understanding why mathematics plays such a central role in physics." Only under the assumption of naturalism. But, if, as I hold, naturalism is irrational, and the only reasonable way of understanding science is an idealistic one (more precisely a mind/mathematics dualism), then your conclusion fails.

      Now for the general ideas : your account of mathematics and its success is much too vague, and fails to relate to the effective contents of how successful is mathematics for physics. You present an abstraction of explanation for an abstraction of a problem. But the difference between mathematical abstractions with their success in science (as abstractions of real problems), and the abstraction of your approach to the problem of the success of mathematics in physics, is that the success of mathematical abstractions depends on their care and success to keep effective logical articulations (as you say, "tether") with the real problems they are abstractions of, but your abstraction of approach to the issue of usefulness of mathematics, is a non-mathematical abstraction that fails at keeping a tether to the reality of this usefulness. If the success of mathematics in physics could be explained as simply as what you describe, there would have been no amazement at this success in the first place.

      Now, of course, it would make little sense to only express this objection in the abstract: to justify it, I need to enter the specifics about how your approach fails to account for the specific success of mathematics. So here are a few specific details of how you fail.

      First, you explain mathematics as "the study of regularities, within regularities, within ..., within regularities of the natural world". In this case, we have a hierarchy of different levels of abstractions, where some concepts are more abstract than others, as they are not directly natural objects but abstract generalities about wide ranges of natural objects, or generalities about generalities about natural objects. When abstract generalities are developed as abstract generalities, they are not themselves the natural objects. That structure of knowledge you describe can only be expected to be useful as a path of reasoning through which we might discover new natural objects by first going up to the abstraction and then down again from the abstraction to concepts of new objects that would be particular objects inside classes described by these abstractions: the new fundamental objects we discover should display this same dependence to mathematical abstract generalities as the objects of our experience do.

      But this is not what we observe. Instead, what we observe is that what we discover as fundamental objects of physical reality are some very abstract mathematical objects themselves. And what I mean here as "abstract mathematical objects", does not look like any generality like a general description of regularities among a range of many particular objects in a naturalistic sense. You may make this confusion because some of the most famous mathematical concepts, such as category theory, precisely look like generalities describing a wide range of particular cases (it is a generality of generalities). But this is not the case for all highly abstract mathematical concepts, and this is not how it goes for the success of mathematical concepts used in physics. In short, "high abstraction" and "generality" are not synonyms.

      And to explain how the concepts of "abstraction" and "generality" differ, I need to take a specific example. I would like you to consider the case of the Dirac equation. This is a particular case of equation of a particular object (electrons or other spin 1/2 massive particles), but nevertheless a very abstract one.

      This equation describes the field of electronic presence as taking values in the space of bispinors of space-time. To say roughly, we can define this space as the sum of 2 spinor spaces (with conjugate types), where a spinor space is a 2-dimensional complex space E such that the space of hermitian forms on E is identified with the tangent space to this point of space-time. Namely, tangent vectors (x,y,z,t) to this point are identified with Hermitian forms on E with matrix

      (t+x yв€'iz)

      (y+iz tв€'x)

      since the determinant of this matrix coincides with the relativistic invariant (t2в€'x2в€'y2в€'z2).

      This is very abstract, but not any kind of "generality of things" like what category theory does by describing regularity classes of particular systems that may go down to objects that look "natural" in a naturalistic sense. Now what is amazing with the success of mathematics is that this spinor space E was found to be "what electrons (and other fermions) are actually made of". Yet its connection to space-time, as described above, is... quite abstract. Spinors are not "made of" space-time connections, since any spinor (element of E) would correspond to a light-like direction of space-time but any physical description by such a direction would fail to fix the phase of this spinor. Concretely, a big problem with a spinor is that its phase is reversed when you apply to it a rotation with angle 2pi.

      So, unless you provide a naturalistic explanation of how an object can be reversed when applying a rotation with angle 2pi, and how such an amazing thing as the Dirac equation can be relevant to physics, I must consider that naturalism fails to account for the mathematical aspects of the physical reality as we observed.

        Dear Matt,

        Good to hear you're a naturalist, not a pragmatist. Or maybe both? :)

        Anyway, I take your argument to be more about mathematical physics than mathematics per se. But even if we restrict ourselves to mathematical physics (which is a branch of mathematics, of course), what about the problem of multiplicity? To follow your argument, the objective world of mathematics is just the physical world, but we know that the physical world admits competing mathematical descriptions (e.g., different formalisms of the same quantum theory: operators, path integrals, geometric approaches, etc.). Why would this multiplicity arise if the mathematical world and the physical world were one?

        Cheers,

        Alexei

          Would you say that another way to say 'regularities within regularities' is 'math theory X surrounds math theory Y'?

            Hi Matt,

            I delayed replying until my essay entry was posted.

            You should be able to find all you want to know about multi-scale variety, on the home page of the New England Complex Systems Institute.

            Bounded rationality was formally developed by Herbert Simon, as a heuristic approach to decision problems. Because Bayes' theorem is also motivated by decision problems -- I think it is quite natural that you would apply Bayesian solutions to problems of physics, and include your philosophy of mathematics in the solution. However:

            In my own journey into knowledge of complex systems, it hit me right between the eyes, years ago, when I read Bar-Yam's seemingly innocuous statement -- "Ashby's law of requisite variety is a theorem in complex systems science." As a mathematician, it did not sit well with me, because 'theorem' has a specialized meaning. And because I knew that Bar-Yam was trained as a physicist I thought it brash and unprovable, even to the extent that I believed I could find a contradiction in the 'theorem.' To my surprise, I found that it really is a theorem, that opens up a whole field of *physical* solutions to problems of information and knowledge. Simon's convention, like yours, is the hierarchical framework. Combining Ashby's law with his own theory of multi-scale variety, though, Bar-Yam provided a compelling argument for laterally distributed information on every scale, This definition of local boundary, and lack of global boundary, convinced me that multi-scale variety is physics independent of any philosophy of physics. That sharp demarcation of philosophy from physics, is, in my opinion, what makes possible a rational correspondence of mathematical model to physical result.

            Anyway, my abstract begins with the words, "Mathematics is not physics." I hope we have a meaningful dialogue over the extreme contrast in our views.

            Best,

            Tom

            That's not a bad way to put it I suppose, but the topology of the situation may be a bit more complicated than one theory being at a lower level than another.

            I am both a naturalist and a pragmatist, and I think that is important for answering your questions.

            Our physical and mathematical theories are both highly constrained by the natural world, but they are not completely determined by them. There is also the constraint that our knowledge derives from a social process and must be represented in a form that is useful to that society (this is the pragmatism part). The fact that there may be several different such representations is therefore not much of a problem for me. I am not saying that the mathematics literally is the physical world.

            I also think you misunderstand me if you think I am talking just about mathematical physics. It could be read that way for sure, but I really intend it as a theory of all of mathematics. I recognize that this is a harder thesis to defend, but I wish to defend it.

            There is a lot of food for thought in your comments, and I don't have time to answer all of them in one go. This is just the first of several replies and I will address the rest of the issues you have raised in due course.

            For now though, I just want to comment on "naturalism". I think it is unfair to criticize the term on the grounds that it does not have a clear and unique meaning. If you look at any similar philosophical term, such as even your favoured "idealism", you will find that they typically refer to a broad church of views that have a main theme in common. The term itself serves as code for this set of views and it may be used when the distinctions between the sub-varieties are not too important for the issues under discussion. Add to this the fact that FQXi essays are supposed to be pitched at a general audience, and I don't think I have been too vague in using this term.

            As far as my stance is concerned, I mean two things by the term "naturalism". The first is that the results of scientific enquiry are not to be ignored when they are relevant to a philosophical enquiry. This is a fairly innocent claim of methodological naturalism, that I think is fairly mainstream in western philosophy. So, for example, an enquiry about consciousness should take into account the results of modern neuroscience. For mathematics the main implication of this brand of naturalism is that, since modern theories of physics use advanced mathematics in an indispensable way, we need to find a theory of mathematics that explains why this is so rather than leaving it as an unexplained miracle.

            However, I do also adopt a stronger version of naturalism, which you may want to brand "materialism", but would more properly be called "physicalism". Here, the idea is that our best guide to what fundamentally exists in the world is physics, so I want a metaphysics that does not posit entities beyond those of physics, or at least one that does so only minimally. Note that, just as with empirical science, the implications of this view are revisable. If a scientist one day discovers reliable evidence for ESP, or is able to reliably detect "mind particles" or some such entity, then the idea that we have to explain everything in terms of what we nowadays call physical entities would have been shown to be incorrect. The point, however, is that this is a matter for science to decide, and we should not go around positing such entities purely for the purposes of our metaphysics.

            As for psychology and the like, physicalism is perfectly consistent with the concept of emergence, of which the emergence of thermodynamics from statistical mechanics is a prime example. The emergence of consciousness, human psychology, sociology, etc. is supposed to be explained in a similar way, but since they are much more complex than thermodynamics, we cannot boil this down to a few simple equations and relationships. Nevertheless, we have no good evidence that they require positing any new fundamental entities beyond those of physics.

            You are right that I do not enter into a defense of naturalism or physicalism in my essay. The starting point is to assume these things as premises and to see what theories of mathematics they are compatible with. That is why "platonism falls afoul of naturalism" is appropriate in the context of this essay. I do think naturalism and physicalism are fairly mainstream positions in philosophy of science, so I don't think it is inappropriate to argue from them. However, I agree with you that, in a broader context, these notions require a defense. Getting into the general debate would take us into deep waters that are probably not too relevant for the philosophy of mathematics specifically. However, it is important for naturalists to come up with a viable theory of mathematics. If we cannot do this, as you seem to think we cannot, then that means that we would have to abandon naturalism, at least for mathematics, so this is an important issue we should debate further.

            Finally, regarding the comparison of my essay to Lee Smolin's, you say I have been vague but I don't think so. Lee thinks that, for physics, we can get away with number, geometry, and maybe a few other things. For him, these things are straightforwardly "real" and the rest of mathematics just a formal game. Lee's views are actually quite close to those of Quine on this issue. Both of them think that it is possible to ring-fence some areas of mathematics as the "physical" ones, and not be too bothered with the rest. I do not agree with this position as I think that more advanced mathematics is truly indispensable for modern physics and it leaves the applicability of such mathematics to physics a total mystery.

            In contrast, for me, there is no distinction to be made between "physical mathematics" and the rest of mathematics. ALL of mathematics is derived from the natural world, and ALL of it is real in exactly the same sense. I hope this is a clear enough statement of my main thesis and how it differs from Lee's. There are, of course, subtleties. I am at least partially a pragmatist about scientific truth, so when I call something "real", what I really mean is that it is useful, indispensable, etc. to the entities who practice science. Therefore, to defend the idea that mathematics is real I have to explain how it is constructed and why it is useful, rather than trying to locate it explicitly in the physical world as a nominalist would. This applies to all of mathematics, including the basic concepts like numbers as well as more advanced branches of mathematics.

            I'll get into some of your more specific criticisms later, but I hope I have at least clarified my main position.

            Matt, you write, " ... I want a metaphysics that does not posit entities beyond those of physics, or at least one that does so only minimally."

            I think this could not be a clearer statement of an anti-rationalist viewpoint. Those entities that are metaphysically real -- such as the moon when no one is looking at it -- is not just minimally real. It is real or it is not.

            " ... when I call something 'real', what I really mean is that it is useful, indispensable, etc. to the entities who practice science."

            What is useful or indispensable about the moon when no one is looking at it, to a scientist or to anyone else?

            Tom

            "I think this could not be a clearer statement of an anti-rationalist viewpoint. Those entities that are metaphysically real -- such as the moon when no one is looking at it -- is not just minimally real. It is real or it is not."

            "Minimally real" is not a terminology I have introduced or at all relevant to what I am saying. The atoms and molecules that make up the moon are real. There is a particular arrangement of those atoms and molecules that we call "the moon". That is also real.

            "What is useful or indispensable about the moon when no one is looking at it, to a scientist or to anyone else?"

            The pragmatist criterion of "usefulness" is very often misunderstood. It is intended in a very broad sense. Theories of the solar system that say that the moon is real are more coherent and tell a more consistent story than those that do not. A scientist who goes around thinking that the moon is real will have a far easier time reasoning about what goes on in the solar system than one who does not. It passes the pragmatic test of "usefulness".

            In fact, I would argue that realist theories are pragmatically preferred in general, as they provide a better explanatory framework than anti-realist theories.

            "There is a particular arrangement of those atoms and molecules that we call 'the moon'. That is also real."

            Physically or metaphysically? If physically real, how does one demonstrate it without disturbing the arrangement?

            "The pragmatist criterion of 'usefulness' is very often misunderstood. It is intended in a very broad sense."

            Then it could mean anything, understood only in the private context of the understander.

            "Theories of the solar system that say that the moon is real are more coherent and tell a more consistent story than those that do not."

            Really? What theory of the solar system says the moon is physically real? I hope you're not thinking of general relativity, where spacetime is physically real and guides the motion of the planets. The planets themselves are metaphysically real objects of the field dynamics, not independently physically real. Or perhaps you are thinking of quantum field theory -- are the particles real, or the field? No field theory, in fact, is dependent on real objects.

            "A scientist who goes around thinking that the moon is real will have a far easier time reasoning about what goes on in the solar system than one who does not. It passes the pragmatic test of 'usefulness'."

            It would, if science were a pragmatic enterprise, rather than a rationalist enterprise.

            "In fact, I would argue that realist theories are pragmatically preferred in general, as they provide a better explanatory framework than anti-realist theories."

            Zeilinger is anti-realist. Do you think he is handicapped by his philosophy? Do you think that he thinks his explanatory framework is inferior to that of a realist?

            5 days later

            Dear Matt Leifer,

            Is it 1. "our universe is nothing but a mathematical structure and that all possible mathematical structures exist in the same sense as our universe."

            or 2. "mathematics is a natural science--just like physics, chemistry, or biology--albeit ... fundamentally a theory about our physical universe and, as such, it should come as no surprise that our fundamental theories of the universe are formulated in terms of mathematics.

            Very interesting argument. I guess one idea that would support your side is that it is hard to imagine people developing things as simple as numbers and counting, if there was nothing to count or organize into classes and groups in the real world, ie. you need something to count to invent the concept of counting...

            Enjoyed your essay, thanks.

            Regards,

            Ed Unverricht