Hi Tim,

Nice essay. I will have to read more about your Theory of Linear Structures at some point. It seems to have some similarity to what Kevin Knuth has done with posets, and with some of the things I'm doing right now with information orders on domains (building on the domain work of Keye Martin -- not sure if you are familiar with any of it).

I think you also hit on something interesting in regard to this idea of counting and how it relates to a physical ontology. One could argue that, even if the universe is entirely continuous, our ability to measure it to arbitrary accuracy is necessarily discrete and thus the integers match up well with that discreteness (which interestingly links back to a previous FQXi essay contest). Just a thought.

Anyway, I nevertheless must admit that I didn't find your argument convincing in general. It seems to miss some subtleties. Perhaps these subtleties are addressed in your larger work on the topic, however. For instance, I disagree with you on a key point: I do think that how different branches of mathematics relate to one another, has a direct bearing on how mathematics relates to the physical universe. How could it not? If you are familiar with category theory or topos theory, think about how such theories describe both mathematics and physics and their inter-relationships.

I had a few minor quibbles as well. In the example you gave of a universe describable entirely via fluid mechanics and dynamics, you would still be faced with the distinction of "something" versus "nothing" which maps quite naturally to 1 and 0 respectively. Integers are an elementary extrapolation from there.

I also am not particularly awed by the fact that results in semi-stable elliptical curves were used to prove Fermat's Last Theorem. While I am not deeply familiar with the details of Wiles' proof, in some sense both elliptic curves and Fermat's Last Theorem deal, on some level, with polynomials. Certainly the connection is not obvious, but neither is it all that shocking, at least to me.

Cheers,

Ian

    Dear Ian,

    Thanks for the comments. Let me try to address some of them.

    There are surface similarities to how one treats discrete spaces or space-time using this formalism and what is done in Causal Set theory (which also uses posets), but the actual details turn out to be quite different. Of course, I did not have the space to go into that here, and it is not even in the book that is out, which deals solely with the math. The second volume will apply the math to physics, and it will be done there.

    I friend of mine pointed out that for another reason there are countable things even in a fluid mechanics continuum: there can be discrete vortices. (There will be problems counting when they merge, but still they can be stable and discrete over long periods.) So the claims about fluid mechanics is too strong.

    I think you misinterpreted the claim about the bearing of different branches of math on one another. Of course that has implications for the connection between mathematics and the world! My point was that if one branch has unproblematic relation to the physics, then any other mathematical structure which connects to the unproblematic one will inherit a comprehensible bearing on physics. In this case, I said that Wigner's problem is solved without remainder. I just wanted to separate puzzlement about why one branch of pure math bears on another from the question of why any math bears on physics.

    The example of Fermat was just illustrative: maybe the connection is not so obscure. Like you, I do not know the details. Take the Moonshine conjectures then: certainly mathematicians were surprised about the connections between group theory and the Fourier expansion there. But if the physics were using the group theory in some obvious way, the purely mathematical connection would make the Fourier expansion relevant to study.

    Cheers,

    Tim

    Dear Prof. Maudlin.

    Despite some interesting ideas, you paper pressuposes what it was supposed to explain. As I see it, there is nothing, absolutely nothing intrinsically mathematical in brute nature. Take for instance number. Given any, any!, amount of objects, no matter how sharply distinct one from another, from a certain perspective, there's no number naturally attached to it independently of a unit determination, or, which is the same, a concept which tells us what is it that we are numbering. So, numbering is a conceptual operation and concepts are creatures of ours. In my paper ("Mathematics, the Oracle of Physics") I approached the question of the applicability of mathematics in science from a transcendental perspective. Since nature "out there" has nothing intrinsically mathematical about it, how come that mathematics has anything to do with our theory of nature? From my point of view, the answer to this question requires showing how by a series of constituting acts a suitable mathematical representation of nature is constituted from brute sensorial data. Once this is done the applicability of mathematics in physics is, as I've argued, just an instance of the applicability of mathematics in mathematics itself (in your paper you explicitly reject this identification). In short, I don't think your perspective is radical enough from a truly philosophical perspective. You take too much for granted and embrace too many idees recues. Thank you! Best! Jairo Jose da Silva

      Dear Jairo Jose de Silva,

      The main contribution in the essay does not concern the application of numbers to physical states, but rather the application of geometrical descriptions. I used enumeration as an example, but it is not the main focus. So the issue of a "unit determination" never arises. Perhaps you do not find the ideas radical enough--one can try to be more radical--but the notion that space-time has an intrinsic geometrical structure is coherent and consistent. Given the right geometrical concepts, one can also see how temporal structure can generate such a geometry.

      The problem with "transcendental" arguments, at least as Kant deployed them, is that they were supposed to explain how we can have various sorts of a priori knowledge. But as it turns out, we just don't have that knowledge. So the transcendental approach does not fit with what we now know.

      Regards,

      Tim Maudlin

      Tim,

      You say that "If there are physical items so constituted as to be solid objects, held together by strong internal forces and resistant to fracture and to amalgamation, then they will be effectively countable"; and that "The relevance of the theory of integers for physics is unproblematic so long as the way that physical items are being counted is conceptually sharp."

      But counting implies making distinctions (1) and counting is necessarily a multi-step procedure, - so who or what is performing the counting procedure?

      Lorraine Ford

      1. "What is a Number?" by Louis H. Kauffman, http://homepages.math.uic.edu/~kauffman/NUM.html

        Dear Lorriane,

        I would not say that a sharply defined enumeration requires any agent or anyone performing anything at all. What is required is the existence of a well-defined map from a physical situation to the integers, and the question is what physical characteristics the situation must have for the map to be well-defined. If I put a bunch of jelly beans in a jar and say there is a definite, exact number of jelly beans in the jar, that is true whether or not anyone ever counts them or goes through any multi-step procedure. It is because there already exists a definite number of jelly beans in the jar that if we want to find out how many there are, it does not matter who does the counting as long as they count correctly, or how they do it. The number they come up with will be the number that is already there before they start counting. If the jelly beans start to melt and merge then (again quite apart from anyone doing anything) the conditions required for a definite number may no longer obtain.

        Regards,

        Tim

        Dear Dr. Maudlin,

        Perhaps you would be willing to say more about the relationship between the geometry of linear structures and the question whether time is asymmetric. I am concerned about the position, presented by Huw Price and others, that there is no intrinsic difference between the directions of past to future and of future to past. It seems to me that a geometry based on the concept of the line might be more helpful on this issue than a geometry based on the concept of the open set. In the first place, a linear-structure geometry could clarify the discussion. We could formulate one very basic issue as the question whether the mathematical structures more accurately corresponding to physical time are directed lines or undirected lines. Furthermore, as you point out, the theory of linear structures would be appropriate for either answer. So, just as with the question of continuity or discreteness, the mathematical language would not add its own weight to the scales when we are trying to investigate a problem about physical facts. As you said in response to an earlier comment, this neutrality is an advantage.

        Best wishes,

        Laurence Hitterdale

          Dear Laurence,

          As I see it, the situation is this. There are three possible positions on the direction of time: 1) there is no directionality to time at all--the direction from this event to the past is physically just like the direction from this event to the future--; 2) there is a directionality, but it is not fundamental but rather should be analyzed in terms of something else (e.g. an entropy gradient). Note that on a view like this, using the entropy gradient to define the direction it comes out analytic that entropy never decreases, since the direction to higher entropy is the forward direction. The "direction of time" can flip around on this view if entropy behaves the right way. This is not what we normally say: we say that the entropy of (say) a gas in a box can go down, and even does go slightly done, even if on average it goes up. 3) The direction of time is a real, physical difference between the time directions, and does not get reduced to or analyzed in terms of anything else. Huw holds 1), I hold 3), and lots of people hold 2) offering different sorts of analyses.

          If you hold 3), then the best one can ask for from mathematical physics is a clear mathematical representation of this directionality. Since we are talking about an intrinsic directionality in space-time geometry, one would need a geometrical language in which directionality can be naturally represented. In standard topology, this is not true. If I ask you to "put a direction" on an open set of points, it is not at all clear what I am asking you to do. But lines, in contrast, are exactly characterized by having only two directions on them. Indicating that these directions are physically different is just a matter of associating one of the two possible directions with the line. This is what can be done using Directed Linear Structures: if the lines have directions, the geometry becomes intrinsically directional. There is no standard topology analog for this at all.

          Now we push further: if space-time has an intrinsically directed geometry, what is the source of the directionality, the directional asymmetry? In Relativity, there is a very natural answer: the directionality is produced by the asymmetric nature of time. Some pairs of events (but not all) can be characterized by an asymmetric temporal earlier/later distinction. And in Relativity (but not classical space-time) that distinction alone is enough to recover almost all of the space-time geometry: everything up to the conformal structure. The whole light-cone structure gets built in, and indeed a complete Directed Linear Structure can be defined. So the picture is that the fundamental asymmetry of time creates a fundamentally directed space-time geometry. And this can be done for both continuous and discrete structures.

          In sum, if you think time is intrinsically asymmetric (unlike Price), standard topology provides no way to easily represent that feature of the geometry and the Theory of Linear Structures does. That is not itself an argument for the directionality. But it is a response to someone who says: "I don't see any time asymmetry in the math!". That, of course, depends on what mathematical language you are using to represent the physical situation. Maybe it is hard to see the directionality in the math because you are using math that does not have a simple way to represent directionality.

          I hope this is some help,

          Cheers,

          Tim

          Hi Tim,

          Re "there already exists a definite number of jelly beans in the jar":

          Where or how does this number exist? Seemingly this number also has a category: "jelly beans in the jar". How is this 2-part (i.e. number and category) informational entity that exists interconnected to physical reality?

          Cheers,

          Lorraine

          HI Lorraine,

          I'm not sure how to take the question "how and when does this number exist?". There is a particular physical situation--jelly beans in a jar--and it is clear where and how that physical situation came to exist. It came into existence when the jelly beans were put in the jar by a certain physical act. The point is that the physical situation so created is, because of its physical character, one in which a particular number can be used to describe the situation. The number itself, the thing being used to as part of a representation of the physical state, is not the sort of the thing that exists anywhere or is created. It is, if you will, a Platonic entity. Wigner does not doubt the existence of such Platonic mathematical entities, he just wonders why some of them would be of any use in describing physical situations. The "interconnection" here is the connection between a representation and a represented object. The representation is mathematical and the object physical. It is not a physical connection between the number and the physical situation, if that is what you have in mind, but a representational connection.

          Framed this way, the only remaining questions is why certain physical situations or entities have a structure that is usefully represented using particular mathematical entities. That is the question I am trying to clarify, particularly for geometrical structure.

          Regards,

          Tim

          If you'll pardon the intrusion ...

          Tim, your jelly beans in a jar made me think of Mandelbrot's question about the length of the coastline of England.

          Measured in uniform units of jelly beans, there would still be no definite answer, no pat number -- and that's even without assuming melting and merging. So I think you are quite right that the number exists independent of the counting units -- because it's scale dependent. No agent required, therefore -- the number is determined by scale of observation, not the observer's choice of measurement units.

          Best,

          Tom

          Dear Tom,

          I see your point about Mandelbrot, but here too we should be careful. If the coastline really were a fractal--which would be a physical feature of it--then there would be nothing that counts as "the length" of it. But if it is not fractal, and becomes smooth at fine enough scale or discrete at fine enough scale, then we could define the "true length" as the limit as scale gets finer and finer. In the discrete case, this bottoms out and in the non-fractal case there is a well-defined limit. So the nature of the dependence on scale is itself something that depends on the physical situation. The fractal would give one extreme sort of case.

          Cheers,

          Tim

          Dear Tim Maudlin,

          When David Joyce commented on a previous essay of mine "it contains some interesting points", I was not sure whether at least he understood my observation that Dedekind replaced Euclid's 1-D notion of number as a measure (or as you are calling it a line?) by the 0-D point at the end of the distance from zero. I understand that it might be no opportune to question the fundamental of point set theory and point set topology. Did you deal with this perhaps historically decisive change?

          Since I read Fraenkel 1923, I am sure to understand Cantor's logical flaw. My strongest additional argument is the indisputable fact that alephs in excess of 1 didn't prove useful.

          Concerning my distinction between Relativity and relativity, see the essay by Phipps. My opinion that there is no preferred point in space but the natural zero of elapsed as well as future time might be too bewildering to those like you.

          Because English is not my mother tongue, I had sometimes difficulties to clearly understand what you meant, e.g. on p. 5 with "sifting humor". On the same page, it would be helpful to find out where footnote 2 refers to and what conjugate points are meant.

          Just an aside concerning Wigner: Von Békésy got a Nobel prize for his claim of a a passive traveling wave in cochlea, the mathematics of which was provided by Lighthill and was indeed unreasonably effective in the sense it was just fitted to measured data. Already Thomas Gold had argued that a passive traveling wave cannot work at all. Later on the cochlear amplifier was found.

          Respectfully,

          Eckard Blumschein

            Dear Eckart Blumenschein,

            I can reposed quickly to the questions about my essay. The term "sifting humor" was used by David Hume, and it means continuing to analyze some concepts even further. Readers familiar with Hume would pick that up, but probably if you had not read the passage in Hume it would sound odd even to a native speaker.

            The conjugate points I have in mind can occur in models of General Relativity (but not Special Relativity) where distinct light-like geodesics intersect more than once. Call two such intersection points A and B. In such a case, each path is light-like even though A and B are the endpoints of two different lines (in my sense). So the criterion for a light-like geodesic that works well in Special Relativity fails in General Relativity. But this can be fixed, because each light-like geodesic can be subdivided into overlapping parts, each of which satisfies the simple definition. So the General Relativistic case can be covered by a simple amendment to the definition.

            Regards,

            Tim Maudlin

            Dear Tim Maudlin,

            Thank you for your quick response. Lee Smolin lost my admiration because up to now he did not even respond to my simple request whether he actually meant "off", what was not understandable to me, or simply "of". I wonder why nobody else admitted not having understood your term "sifting humor".

            We all make mistakes. Misspellings of my name don't matter; here is no risk of confusion.

            I am looking forward learning from your criticism of my admittedly uncommon arguments.

            Regards,

            Eckard Blumschein

            Dear Tim:

            This is a fantastic essay. And it is very well written.

            As we all know, mathematics has been very effective in physics. Its weaknesses to date in modelling physical reality have been twofold:

            1. Using open set theory, it does not model that time has an order (whether we interpret this as an order in forward time or an order in backward time);

            2. It does not model which directed order (forward order or backward order) corresponds to the observed Arrow of Time.

            Your Theory of Linear Structures addresses point 1 - and is therefore important. However, I do not believe it can address the second point. In particular, the initial end point of a line could represent a past instant of time and the final end point of the line could represent a future instant of time - but equally the initial end point of a line could represent a future instant of time and the final end point of the line could represent a past instant of time. The mathematics cannot differentiate between these two interpretations. So, the Theory of Linear Structures, while it can encode an order, will not I believe be able to encode which directed order of a line corresponds to the observed Arrow of Time. Instead, we would have to impose the direction of time from outside the mathematics on the solution - as we do now for open set theory.

            What this would mean is that the Theory of Linear Structures - although important - will produce time-symmetric theories, as open set theory does now. We will unfortunately not be able to use it, for example, to have a mathematical theory of evolution. Nevertheless, your work is very good.

            If you are interested, in my essay I explain more generally why mathematics cannot, in principle, model the Arrow of Time.

            Thank you again for some great research and a clear essay.

            Kind regards

            Spencer Scoular

              Dear Spencer,

              Thanks for the careful reading. I don't think we disagree on this much: there is a real, physical distinction between the two time directions and it would be useful to have a way to describe geometries that include such a fundamental asymmetry. That is the part my language does. Then we ask a further question: can we give a more profound account of the physical nature of the asymmetry? This could turn out two ways: it could be a fundamental physical structure, and so not admit of further analysis, or there may be a deeper analysis possible. I am open to both possibilities, and it sounds like your work addresses the second.

              My hope is that this mathematical structure can be of use for many different projects. Perhaps yours is one.

              Cheers,

              Tim

              Dear Tim Maudlin,

              I know from your work that you have a strong acquaintance withh Bell's work (B). I arrived at Bell/CHSH inequality from my investigation of Kochen-Specker theorem for multiple qubits mainly through Mermin' treatise (my ref. [19]). At some stage, I observed that the commutation diagram for a set of four observables involved in the violation of the inequality is just a square/quadrangle.

              Hence my attempt to deepen the subject. My work on KS in dimensions 4, 8 and 16 (two, three and four qubits) is in 1204.4275 (quant-ph) [Eur. Phys. J. Plus 127,86 (2012)] where I also mention a paper of P.K. Aravind on BKS.

              My 2013 FQXi essay [also 1310.4267 (quant-ph)] provides the details you ask for. The inequality in p. 4 of my present essay is that of Peres's book [(6.30, p. 164 of Quantum Theory: Concepts and Methods, Kluwer, 1995]. Replacing the dichotomic variables s_i by the appropriate (i.e. commuting like a square) two-qubit operators (or n-qubit operators) that have dichotomic eigenvalues +/-1) as in Peres, p. 174, the norm of C equals 2v2. With two-qubits, there are 90 distinct squares/violations, some involve entangled pairs of operators, others no (as in my example of Fig. 2a). B or KS is not a matter of entanglement but of contexts (compatible observables) as already recognized by many authors. Here I don't refer to an interpretation of QM but to a strict application of its domain of action.

              Of course on can go ahead and try to discover a realm for squares and other finite geometries relevant for BKS as I started to do in the 2010 FQXi essay and my subsequent work. I have found that the application of Grothendieck's dessins d'enfants is very promising in this respect. I have been quite surprised that stabilizing a particular square from the two-generator index 4 free group is an instance of the smallest moonshine group (p. 5) whose structure amounts to that of the Baby Monster group.

              I hope that it clarifies a bit what I wrote. I am currently working at your own ambitious essay and I intend to give you some comments in the coming days.

              All the best,

              Michel