This quarter I am taking a seminar purportedly on “new wave” philosophy of mathematics. While we have talked about a lot of interesting things, not much of it has seemed particularly “new wave”. We have read some Friedman on Carnap, and a book by Parsons that, while it is inarguably new, seems to be very much concerned with classical issues of mathematical ontology.

As a result, I have periodically taken up reading things on my own that seem more modern. In particular, last night I read a chapter from The Architecture of Modern Mathematics by Jean-Pierre Marquis entitled “A Path to the Epistemology of Mathematics: Homotopy Theory”. This, similarly to the mathematical explanation stuff I wrote about (in a way) yesterday, is fundamentally different than the classical concerns in the philosophy of mathematics.

Although philosophy of mathematics has, in the past, been very occupied with mathematical epistemology, most of the focus has been on issues of mathematical intuition. This chapter, however, is not concerned with that. The focus is trying to draw a parallel between general scientific technologies and their emergence and systematic mathematical technologies. To illustrate the point, the paper takes a look at algebraic topology generally and homotopy theory specifically.

Homotopy theory is a part of algebraic topology concerned with equivalences of paths and path generalizations. By a path in a topological space , all that is meant is a continuous function . We can then consider two paths and to be equivalent when there is a continuous map such that , . This generalizes, allowing the replacement of by and by .

The details are largely irrelevant to the main point except as an example. The main thing I took away from the paper is the importance “for many mathematical notions, to understand that notion, it is irrelevant to specify what it is ‘made of’, or its underlying ‘ontology’ but rather what it is used for: when, why and how.” (245-6) The case of homotopy turns out to be an excellent example of this. Things like fibrations and spectral sequences, while perhaps interesting in their own right, are really only understood when one grasps how, when, and why to employ them as tools for calculating homotopy. Furthermore, homotopy theory itself, while interesting, is best understood in terms of the goal of classifying topological spaces. That is, while the “axioms of homotopy theory” are general enough to apply to many categories, and that makes them intrinsically interesting, their use for developing theories for the classification of spaces is far more central to really understanding them.

In general, it seems like if I am going to continue in the philosophy of mathematics, it will be with regard for things like this. Focusing on mathematical practice with deep examples, answering interesting questions and probably not being concerned with ontology or other seemingly dead-end classical questions.

Since the NSF finally got around to releasing the last GRFP decisions today¹, I thought I would post a bit from my research proposal. I haven’t thought about this stuff for some time, but reading it again is starting to make me want to actually undertake some of the work.

Research Proposal Background Sketch

Science can generally be seen as undertaking the task of explaining empirical facts — answering questions concerning why things are the way they are. Although mathematics is similar to the natural sciences in many ways, there are substantial differences that have caused problems when attempting to apply theories of explanation from the philosophy of science to mathematical practice.

Although a number of papers have been published on the notion of explanation in mathematics, nothing has been satisfactorily decided. Suggestions have included the unification theory of Kitcher (1981), the use of characteristic properties that provide opportunities for proof deformation in Steiner (1978a), and van Fraassen’s theory of why-question logic as discussed by Sandborg (1998). The examination of various case studies, however, have led Hafner and Mancosu (2008) to successfully argue against Kitcher (1981), and Resnik and Kushner (1987) have successfully argued against Steiner.

Despite the failure of each proposed theory of mathematical explanation thus far, searching for an overriding theory is still fruitful. In the naturalist program, for example, mathematical explanation could be construed as a theoretical virtue of mathematics. For instance, perhaps it should be considered when attempting to adjudicate the adoption of new axioms in set theory. (see Maddy 1997; 2007; Mancosu 2008) More importantly, it is widely accepted that the study of actual mathematical practice is vital to improving understanding of the nature of mathematics.

The study of mathematical explanation and the nature of mathematics has definite connections to general philosophy, history, and practice of mathematics. For instance, a better understanding of mathematical practice may help explain why, when set theory was inconsistent, things didn’t “explode.” Such answers are intrinsically interesting, and they might also prove useful should similar problems reoccur. Moreover, practicing mathematicians might reconsider the style in which they present results. This, in turn, may allow for quicker and more efficient distribution of applicable developments to other fields.

Additionally, mathematical explanation is connected to scientific explanation in general. Steiner (1978b) considers the relationship between mathematical explanations of scientific facts and mathematical explanations of mathematical facts to be alike in methodology. Although mathematical practice and general scientific practice appear substantially different, a general theory of mathematical explanation may throw light on explanation in other sciences and on explanation in many other corners of academia.

References

Hafner, J. and Mancosu, P. (2008). “Beyond Unification” in The Philosophy of Mathematical Practice 151-178. Oxford University Press.

Kitcher, P. (1981). “Explanatory Unification” in Philosophy of Science 48(4):507-531.

Maddy, P. (1997). Natrualism in Mathematics. Oxford University Press.

______ (2007). Second Philosophy: A Naturalistic Method. Oxford University Press.

Resnik, M. D. and Kushner, D. (1987). “Explanation, independence, and realism in mathematics”. The British Journal for the Philosophy of Science, 38(2):141-158.

Sandborg, D. (1998). “Mathematical explanation and the theory of why-questions”. The British Journal for the Philosophy of Science, 49(4):603-624.

Steiner, M. (1978a). “Mathematical explanation”. Philos. Stud., 34(2):135-151.

______ (1978b). “Mathematical explanation and scientific knowledge”. Nos, 12(1):17-28.

¹ I ended up getting an Honorable Mention, for the record.

I have started reading a book on Measurement theory, so I’m going to try to write a little bit about the little bit I understand of it.

What is Measurement Theory

In general, measurement theory deals with relating physical things and mathematical things meaningfully. That is, it is a theory of taking a physical relational structure and, via “measurement”, attaching a corresponding mathematical relational structure.

The book I am reading focuses on isomorphisms of structures exclusively. This in general bothers me, because a lot of the examples are isomorphisms of physical things with real continua without justification of why the physical stuff is actually continuous and not just dense. However, I am assured that similar results can be obtained with homomorphisms in place of isomorphisms, allowing for the math to come out substantially more messy.

Why should we care about it?

My personal take on this goes back to my view on conventionality in general. A theory such as this can be useful in explaining the details of something like¹ the correspondence rules of Reichenbach or the conventionality of Schlick with respect to the connections between mathematics and physics.

Somewhat separately, measurement theory can illuminate why we can² bring to bear totally unrelated mathematical tools to a physical problem — perhaps from quite disparate domains of mathematics — while still getting a reasonable result at the end of the calculation or transformation.

There are surely other reasons, but I am only at the beginning of the text that is introducing me to the subject. I will write more as I get further along.

¹ Read: perhaps a caricature of the ideas of

² Or, perhaps more interestingly, why we can’t in some situations.

Recently, I have been reading Charles Parsons’s Mathematical Thought and its Objects for a seminar¹. As a result, I have been brought back to seeing how silly caring about mathematical ontology can be. Presumably we can all agree that mathematicians talk about sets as “objects” standing in the relation “membership” to other sets. But Parsons feels the need to question whether there is something more to that to the ontology of set theory. Granted, he is attempting to defend a sort of structuralism about mathematics, but even that seems strange.

Regardless of what mathematical entities “actually are”, it seems that the more important questions are methodological. Mathematicians are not going to stop using certain ways of speaking because some philosophers finally agreed, for instance, that mathematical objects really don’t exist. Rather, more interesting questions involve how transformation occurs within mathematical practice and the standards for the mathematical community. This can come in many forms, including the naturalism/Second Philosophy of Penelope Maddy², or more controversially in the form of conventionalism in the spirit of Carnap or Poincare.

Regardless of what I tend to consider important, however, it is probably a good idea to get used to thinking about mathematical ontology and epistemology, since presumably there is significant potential for me to get ambushed with questions about it later. So back to reading and toting around a red “M“.

¹ Parsons, Charles (2008). Mathematical Thought and its Objects. Cambridge University Press.

² Maddy, Penelope (2007). Second Philosophy: A Naturalistic Method. Oxford University Press.

So today's post is going to be quite short, but it is connected to the posts I made late last week. In general, I want to ponder what happens in the case of a failure of physical theory.

Let us consider the following thought experiment: Physicists are testing some hypothesis about the structure of space. Presumably this is possible, though I don't actually know. The current best physical theory suggests space is Lobachevskian (that is, is non-Euclidean with constant negative curvature -- from what I recall this is what general relativity suggests, though again, I am not an expert; at any rate, it shouldn't matter). However, the test does not conform to predictions. Furthermore, suppose many other tests also fail to conform to the theory. Now what should happen?

According to Quine, the whole of scientific knowledge goes through the "tribunal of experience" as a whole. As such, it is surely possible to say that Lobachevskian geometry, or some other part of mathematics, is false and should be revised. This seems profoundly strange. Another option is to say that the mathematical model is a bad model of the theory of space; it is not true or false, just performs poorly. Perhaps this way is open to Quine. It seems natural that Quine would suggest that the physical theory of science should be altered to indicate that space is not Lobachevskian. Is that the same as just saying the model was bad? Was there really any chance of mathematics being altered based on physical experience?

It is somewhat harder to imagine what Carnap would say in the same situation. At the least, he would say that the customary interpretation of the geometrical calculus was not a true interpretation. As I believe I have outlined before, this seems strange in itself. I think it is better to say that the geometrical calculi are interpreted analytically, and that it is the correspondence map from the world to the interpretation that is no good. (By correspondence map I mean something like the map(s) that allow for the counting of objects despite the fact that the arithmetical calculus is interpreted analytically within second/third(?)-order logic.) That is, trying to make paths of beams of light correspond to lines, etc. is not a good model of the universe within the geometrical calculus being considered. If it is desirable that the path of a beam of light indeed correspond to a line, then perhaps an interpretation of a different geometrical calculus should be used instead. On the other hand, if it is desirable to continue considering Lobachevskian geometry, a different correspondence should be considered. Granted, this is not really what Carnap thinks, but it seems to me to provide a more consistent theory within which to work. All of mathematics stays analytic L-rules, whereas the correspondence maps and other aspects of modeling are synthetic P-rules.

Yesterday I talked some about Carnap on the synthetic nature of geometry. To recap, geometry is synthetic because the truth of the sentences of the geometrical calculus under its customary interpretation depends on empirical facts. Arithmetic, on the other hand, is analytic according to Carnap, for similar reasons. The customary interpretation of Peano arithmetic is customarily reducible just to a higher-order logic (second or third order), which in turn in analytic. Therefore, the truth of sentences of arithmetic under the customary interpretation of the calculus only depends on logic, making it analytic.

Reflecting on these differences, it seems there is nothing substantially different between the two cases. It is just a matter of the "choice" of customary interpretation that makes arithmetic analytic and geometry synthetic. If, for instance, a different standard interpretation of geometry in terms of modern analytic geometry could be established, geometry should turn to being analytic instead. Furthermore, if the interpretation of arithmetical sentences rested on apples or some other physical object, it seems that it would be considered synthetic.

There is, however, an important difference between arithmetic and geometry that might lead to certain natural interpretations. Geometry is the study of different classes of structures. There are Euclidean, hyperbolic, and elliptical geometries that are mutually exclusive, but also relatively consistent. Similarly to group theory, the point is to develop something of an if-then-ism about geometrical structures. That is, proofs in geometry seem to work something like "If the space is Euclidean, then the sum of the interior angles of a triangle is 180-degrees." Without that antecedent, the statement "The sum of the interior angles of a triangle is 180-degrees." seems like it should not have a truth value, just like the fact that "There is an element of order 7." in group theory doesn't make sense without an antecedent condition of what group or class of groups is intended to be discussed. The same concerns don't apply to arithmetic. There is only one object under consideration, not many as in geometry or group theory. Therefore, the sentences of arithmetic do not need a proper antecedent to be meaningful.

The standard interpretation of geometry, then, seeks to pinpoint a certain geometrical structure to prepend as an antecedent to statements that one might wish to make. That is, it fixes what the basic terms mean and then tries to decide which type of geometrical space, when added as a hypothesis, best makes geometrical statements coincide with empirical data.

We now seem to be left with the question of why sort of argument from geometry doesn't apply to other branches of mathematics. That is, why isn't there a physical group theory and a mathematical group theory? Why isn't there a physical theory of complex systems and a mathematical one?

This post will be in two, totally unrelated parts. The first part is a bit of a rant, and then the second part will concern more about the Carnap problem I posted about last time.

Random Things

So this might not actually be totally unrelated (cf. the age-old view that everything is like everything else somehow), but it is at least not intended to be related to the next part of the post. One thing I have realised lately is that I should be bringing a big, red, "M" placard to one of my seminars if not for the rest of the quarter then at least for the next several weeks. We are getting in to reading Parsons' Mathematical Thought and its Objects, which, at least at the beginning, is highly metaphysical. I have determined that metaphysics is one of those things that I most commonly fall asleep reading, so this should prove interesting.

With that said, now for something completely different...

More on Carnap - Dedution vs. Description

Important to figuring out whether and how arithmetic and geometry are different for Carnap is determining how he characterises deduction vs. description. First, I will start with geometry and description, and then move back to arithmetic.

The crux of geometry being synthetic for Carnap is that the customary interpretation of the calculus in physical. That is, "we may think of a point as a place in the space of nature; straight lines may be characterized by reference to light rays in a vacuum or to stretched threads... Thus the specific signs of a geometrical calculus are interpreted as descriptive signs."[1, section 21] However, this same thing does not carry over to arithmetical calculi. Carnap thinks that "the specific signs of a mathematical calculus are interpreted as logical signs, even if they occur in descriptive factual sentences stating the results of counting or measuring."[1, section 21] Carnap also sees that the geometrical calculus can be interpreted logically as analytic geometry.

However, physical geometry is still different from mathematical geometry. Physicists, he claims, still ask which (Euclidean or non-Euclidean) geometrical structure space has. This seems like a strange question to me in general. It seems to hold the potential interpretation of the geometrical terms fixed and asks which axiom system makes this interpretation best correspond to observation. Carnap writes, "When an interpretation of the specific signs of is established... then each of the calculi yields a physical geometry as a theory with factual content." [1, section 22] The interpretation itself is up to choice.

It seems strange to consider geometry to be synthetic and arithmetic analytic just on the basis of the choice of customary interpretation. Tomorrow I should probably write about whether arithmetic could be shown to be descriptive in the same way geometry is, or whether there is something special about it such that it can only ever be deductive.

References

[1] Carnap, Rudolf. Foundations of Logic and Mathematics. Foundations of the Unity of Science, vol 1, no 3.

Since I am preparing to write a paper about Carnap and his distinction between the natures of analysis and arithmetic on the one hand and geometry on the other, I thought I would write some preliminary thoughts here.

First, it is important to understand the origin of this distinction. Carnap sees arithmetic and analysis to be primarily deductive in nature. That is, they are used as tools to derive statements from premises. They are entirely L-rules. However, geometry is substantially different. Geometry itself can be broken up into a geometrical calculus and an interpreted system. As far as the calculus is concerned, geometry is just like analysis and arithmetic. However, interpreted geometry is not purely deductive any more, but instead it is descriptive. It is descriptive in the sense that it intends to describe aspects of experience, and thus, its rules become P-rules.

With that as a sketch of the background, let me go into some of my concerns. Carnap seems to think that we are wedded to the standard interpretation of the undefined terms of geometry. That is, "point" as space-time point, "line" as path of a beam of light, etc. Under this interpretation, then, he claims there is a correct geometry of space. Originally, I had not paid attention to the "under the standard interpretation" fact that was lurking in the background. As a result, I was very unhappy about that claim, thinking that the way we interpret "point", "line", and "plane" was a free choice. Therefore, with sufficiently clever choice of those and sufficient willingness to have the math and theory turn out very complex and ugly to work with, we could describe space with any geometry we wished.

With that worry out of the way, things are just peachy, right? Well, not so much. The other worry I have is the implications of this for the rest of mathematics. Clearly, the geometrical and arithmetical calculi, when uninterpreted, are analytic and without content in Carnap's sense. However, interpreting those calculi comes with mixed results. Geometry becomes synthetic, whereas arithmetic stays analytic. This is where the point of confusion comes: what is so special about arithmetic that doesn't force it to have the same fate as geometry?

One possible explanation I see at this point is that interpretations of arithmetic are still not intended to describe. If the purpose of the interpreted enterprise is primary, then this could explain why geometry and arithmetic are so different. Another option is something like any possible interpretation of the arithmetical calculus is essentially the same, lending no information outside of its deductive power. This somehow seems more plausible, but I will have to think about it more.