This is the last of my posts on math books for a while (following this one and this one). The last one was on John Stillwell's number theory book. This one is on his Four Pillars of Geometry. This is, as the title suggests, an introduction to geometry, from Euclid through some non-Euclidean geometries. There is, in my opinion, a lot to recommend this book. It begins with Euclid, moving to coordinate planes and vectors, then to perspective, projective geometry, and transformations, and closes with hyperbolic geometry (the thing most commonly associated with the phrase "non-Euclidean geometry" if I'm not mistaken.). The sections on Euclid are quite nice, with lots of exercises and constructions to get you used to doing things his way. There is also a decent explanation about how to do arithmetic with constructions of line segments instead of numbers, which is generalized to work in projective geometry. Unfortunately, there is no proof of the infinitude of primes done entirely in line segments, which, I am told, was how the original was done. There is a lot of discussion of Pappas's theorem and Desargues's theorem in both old Euclidean geometry (plane geometry, I think it is called) and projective geometry and how the two theorems relate. The chapter on transformations explains some of Klein's Erlanger Program (geometry is the study of invariants of groups of transformations) and gives a good sense of what transformations are for various geometries. There are lots of (glorious) diagrams (yay for diagrams) in addition to there being a good selection of exercises. Also, as David Corfield of n-category Cafe likes, there is a decently cohesive narrative element connecting each chapter, giving you the "why"s as well as the "what"s of the proofs and chapters, including a couple of excursions into art to demonstrate the evolution of perspectival ideas. One of the narrative sections prompted an earlier post on the history of the parallel postulate.

Possibly the lowest point of the book (for me) is the discussion of quaternions. They weren't that motivated compared to vectors, which they followed. I don't have any other big complaints though.

In summary, I'd recommend this as an introduction to geometry. I'm not sure what else one would want from an introductory book apart from more stuff on hyperbolic geometry. The book does a very good job of going through conceptual relations in geometry.

Geometry is a bit of a hot topic around these parts. Both Pitt and CMU had dissertations recently on geometry and there was a class offered at Pitt last fall on philosophical issues in geometry. I don't think I'll be joining the bandwagon, but there is certainly philosophical mileage to be found in geometry.

## Tuesday, July 31, 2007

### Philosophers love lines

Posted by Shawn at 10:20 PM 0 comments Links to this post

### Today's mystery

How did the Vienna circle read the Tractatus and come up with what they did?

I've managed to get myself into the position where I can't see how they did that after doing fairly in-depth readings of the book like they were supposed to have done. It seems like they cherry picked bits that sounded good and threw out everything else, which is most of the book.

Posted by Shawn at 10:18 PM 5 comments Links to this post

Labels: Wittgenstein

## Saturday, July 28, 2007

### Philosophers like numbers

A while back I wrote about how I was working through a few yellow UTM books to learn a bit of math that I didn't know. I got sidetracked and didn't write the follow ups that I wanted to. I'm being driven by procrastination, so it seemed like a great time to write. The next book I want to mention is John Stillwell's Elements of Number Theory. I liked his book on geometry, which I'll talk about soon, so I thought I'd try his number theory book. I feel like I know a little bit about natural numbers after all the time spent playing around in Peano arithmetic. This turned out not to be terribly true. The book covers a fair amount. It starts with the natural numbers, integers and diophantine equations. The following chapters are on factorization, the Euclidean algorithm, congruence arithmetic, the application of factorization to cryptology, gaussian integers (complex numbers a+bi with both a,b as integers), the Pell equation, quadratic integers, and the four square theorem. There are a few more chapters on more algebraic topics like rings and ideals, which seem like they would be very neat. I didn't finish this book because I got frustrated with it. There are a decent number of topics in the book which the author relates to the others very well. There is a good bit of narrative connecting the different parts of the book together. This nice feature is unfortunately combined with few proofs and a dearth of good exercises after the first chapter. It seemed like a book that would be fun to go through once you already knew some number theory, but as a book to learn number theory, it falls short. The proofs and exercises are sparser than one would expect. Lame. On another note, Stillwell seems to like quaternions, which are an extension of the complex numbers. From what I gather, quaternions are a relatively obscure domain of math that aren't used much. Both this book and Stillwell's geometry book have at least one section on them. I will hazard a guess that his other books mention them as well. This is just a little curiosity I wanted to mention. It is kind of cute.

In summary, I'd recommend avoiding this book. If anyone has a recommendation for a good number theory book, I'm all ears.

Posted by Shawn at 3:26 PM 4 comments Links to this post

## Friday, July 27, 2007

### Why is this okay?

There was a recent article in the New York Times called "Study says obesity can be contagious". The article is about a recent article in the New England Journal of Medicine that says your chances of gaining/losing weight increases as close friends gain/lose weight. I'm not sure if it is just the reporting or the actual article, but there doesn't seem to be a hint indicating that it is just correlation. There seems to be a causal mechanism being posited. Here are a couple of choice quotes:

"Obesity can spread from person to person, much like a virus, researchers are reporting today."

"The investigators say their findings can help explain why Americans have become fatter in recent years — each person who became obese was likely to drag some friends with them."

"Proximity did not seem to matter: the influence of the friend remained even if the friend was hundreds of miles away."

I think the best comment upon this is the one that closes the article in which a doctor at Cambridge says that good science is repeatable while this study is unique.

There are probably several things about this to discuss, of which I am competent in more or less none. I don't know enough about statistics and statistical methodology to comment. There seem to be causal mechanisms posited wildly. I hope this gets discussed somewhere by someone more knowledgable. But, there is a general question that I have based on ignorance. Biology, psychology, and physics get a lot of play in the philosophy of science world. I have a book on the philosophy of linguistics, but I never see anything else about it. Is there much of a sub-discipline of philosophy of medicine (the assumption being that medicine is a science)? There seems to be so much work done in medical research that seems possibly philosophically relevant, but doesn't get a lot of attention even though (because?) it is in the public eye.

Posted by Shawn at 10:19 PM 7 comments Links to this post

## Thursday, July 26, 2007

### McDowell and the unintuitive

From the end of the fifth lecture of Mind and World:

"It is common for philosophers to think they can dismiss Evans's position, without attention to the wider context I have placed it in, on the ground that they find its implications counter-intuitive. This just reveals the depressing extent to which his ground-breaking work has not been understood. That such work can be so little appreciated is a mark of degeneracy in our philosophical culture."

Granted, the quote goes beyond the title, but the end of that paragraph is too brilliant to ignore.

Posted by Shawn at 12:01 PM 0 comments Links to this post

## Friday, July 20, 2007

### Rearticulating Reasons: Asymmetric terms, the arguing

In chapter 4 of Articulating Reasons, Brandom argues that singular terms and predicates have a certain sort of inferential behavior. Predicate isn't quite the correct label. Rather, he concerns himself with sentence frames, which include predicates (Fx) but also include predicates made of, e.g., logically complex sentences that are turned into predicates by removing an argument (from Fa->Pa, to Fx->Px) Singular terms are symmetrically substituted for and sentence frames are asymmetrically substituted in. There are three views to argue against: both are symmetric, terms are asymmetric while frames are symmetric, and both asymmetric. The first option is dismissed as wrong since both being symmetric would be structurally deficient. The other two are argued against together. The strategy is to show that terms can't behave asymmetrically.

The argument begins by noting that if the inference from Pa to Pb, for terms a and b, is good, then the inference from Pb to Pa will not be good generally. If there is a way to construct a sentence P'x, for arbitrary sentence Px, such that P'x is inferentially complementary to Px, then this will show that there is no way to characterize the asymmetric terms. P'x is inferentially complementary to Px if when the inference from Pa to Pb is good but not conversely, the inference from P'b to P'a is good but not conversely. Such complements can be constructed given either conditional or negation. If the language has a conditional, then P'x is Px->S for some sentence S. If the language has negation, then P'x is ~Px. Now, given any terms with a inferentially stronger than b, and that the inference from Pa to Pb is good but not conversely, and that the language under consideration has negation or conditionals, a complementary P'x can be constructed so that the inference from P'b to P'a is good but not conversely. But, a was inferentially stronger than b, so the inference from P'a to P'b should be good. This means that the inferential strength ordering breaks down, so the terms can't be asymmetric given certain expressive resources.

The argument depends on the notion of asymmetry Brandom uses. One way of understanding the asymmetry of substitution is by taking it to be that if you can infer from a to b in a frame, then generally you cannot infer from b to a in that frame and if you can infer from b to a, then you can't from a to b. Brandom's idea of asymmetry is different. He seems to use a partial ordering of inferential strength of terms such that if for all frames, if you can infer from a to b, then you can't infer from b to a. I'll label the former weakly asymmetric and the latter strongly asymmetric.

The argument rests on setting up inferentially complementary sentence frames, but the inferential behavior of these is weakly be asymmetric. The asymmetry is simply reversed from the normal direction; in terms of the example I've been using, instead of being asymmetric going from a to b, it is asymmetric going from b to a. Weak asymmetry is not eliminated. This does contradict strong asymmetry though. In terms of an ordering, the complementary frames reverse the ordering.

For the following point the ordering of the terms in the non-complementary frame setting will have positive polarity and the ordering in the complementary frame setting will have negative polarity. For the argument to work, it needs strong asymmetry as a premise, so that the positive polarity ordering is preserved in the negative polarity setting; in other words, the positive polarity ordering is the only ordering. Since the complementary frames are constructed using some bits of logical vocabulary, it is possible to discriminate them syntactically. It looks like Brandom tries to get around this by introducing new syntactically simple signs for the complementary sentences, defined using the conditional or negation. I think this is smoke and mirrors. The crux of the argument is the use of strong asymmetry rather than weak. I find strong asymmetry a little bit weird. Since it is stronger than weak asymmetry (not just in title), Brandom should supply some argument or explanation why strong asymmetry is a property of term substitution. We can always pick out the frames in which the polarity of the ordering has been reversed, but the strong asymmetry's appeal seems to rely on not being able to do that. The weak point of Brandom's argument seems to be the strong asymmetry premise. Given that this argument is the linchpin of a foundational chapter of Articulating Reasons, this looks bad for Brandom.

Posted by Shawn at 11:32 AM 2 comments Links to this post

## Thursday, July 19, 2007

### Geometry and intuition

Here is a story I've heard from mathematicians in a few different contexts. I will quote it from Stilwell's text Four Pillars of Geometry.

"The first theorems of hyperbolic geometry were derived by the Italian Jesuit Girolamo Saccheri in an attempt to prove the parallel axiom. In his 1773 book, Euclides ab omni naevo vindicatus (Euclid cleared of every flaw), Saccheri assumed the non-Euclidean parallel hypothesis, and sought a contradiction. What he found were asymptotic lines: lines that do not meet but approach each other arbitrarily closely. This discovery was curious, and more curious at infinity, where Saccheri claimed the asymptotic lines would meet and have a common perpendicular. Finding this 'repugnant to the nature of a straight line,' he declared victory for Euclid."

I've heard this story a few times, and each time it was to illustrate the importance of deriving a contradiction in a reductio, not just stopping when things get unintuitive or "repugnant". I haven't seen the original book, so I don't know if Saccheri says that his conclusions contradict the nature of a straight line, in which case he would have misunderstood what was happening to the notion of line under the new assumption. However, if he thinks that the repugnance is enough for a reductio, he is wrong.

(Apparently there is a lot of literature on intuitions. I've read none of it. It could very well discuss this story in excruciating detail.)

It seems, just from my own reading, that the charge of having unintuitive consequences is often leveled against various theses. I don't have a nice example off-hand, but I'm doubtful that it will be terribly hard to track one down. (I think I used the phrase in the paper I read at UT Austin, but I am pretty sure I have seen real philosophers use it.) It is never clear to me what the force of that charge is. If it is meant in the sense that Saccheri supposedly used it, then that seems to be bad. If it is meant ad hominem, then it is fairly easy to ignore. There are certainly at least some areas of philosophy, logic at least, where the charge of unintuitive consequences is fairly meaningless. Are there any areas in which saying that something has unintuitive consequences is particularly damning? There probably are, but I'm not sure what exactly.

Posted by Shawn at 6:46 PM 6 comments Links to this post

## Thursday, July 12, 2007

### An elementary question

Why is the use/mention distinction important? As far as I can tell, it goes back to Quine, who was very big on it. Some people since him have picked up on it and used it to argue for various things, such as Harman's sharp distinction between inference and implication. I have a vague sense that there is a connection between it and Quine's extensionalism. Both are used as criticisms of modal logic. Modal logic was, according to him, conceived in the sin of use/mention confusion. But, modal logic has been vindicated. What has the use/mention distinction given us? Is it just an artifact of curmudgeonly Quinean philosophy? I currently have no idea.

On a related note, I had an odd experience recently when I realized that my entire undergrad logic education, including a couple modal logic courses in the Amsterdam style, never once touched upon use/mention.

Posted by Shawn at 9:24 PM 4 comments Links to this post

Labels: logic

## Wednesday, July 11, 2007

### A Very Pittsburgh Summer

This summer I'm doing reading groups with some of the other grad students on Brandom's Articulating Reasons and McDowell's Mind and World. I've recently started reading Mark Wilson's excellent Wandering Significance, which I hope to start posting about soon. I've been reading several papers on Wittgenstein and Frege by Tom Ricketts. Maybe I can read Belnap and Gupta's Revision Theory of Truth and get a couple more Pitt philosophers under my belt. To top it off I should probably just eat Sellars's Empiricism and the Philosophy of Mind, the edition with Brandom's study guide of course.

Posted by Shawn at 9:05 PM 0 comments Links to this post

### I sense bias...

On this reading list I came across the following line that annoyed me:

"For a more advanced, controversial and partly historical (but nonetheless very clear) treatment, see John Etchemendy..."

I whole-heartedly agree that Etchemendy's books is somewhat advanced, controversial, partly historical, and very clear, but, I see no need for the snide "historical BUT clear." Clarity and historical orientation are independent.

Posted by Shawn at 11:38 AM 0 comments Links to this post

## Tuesday, July 10, 2007

### Rearticulating Reasons: Symmetric singular terms

In chapter 4 of Articulating Reasons, Brandom argues that singular terms must exhibit a certain sort of inferential behavior, namely symmetric. He argues that asymmetric substitution inferences for terms are impossible. I will present that argument in detail soon in another post, but I wanted to comment on an odd thing about the conclusion. Assuming the argument works, we find that if the language has certain expressive resources, namely either a conditional or a negation, then terms must lisence symmetric substitution. That is a fairly minimal condition, but it is still a non-trivial condition. If we have an impoverished langauge without either of those bits of logical vocabulary, the argument doesn't get off the ground. Why does Brandom try to draw the stronger conclusion that singular terms must behave symmetrically with respect to substitution? Is there supposed to be something about inferentially articulated languages that require them to have at least one of those bits of logical vocabulary? Based on what he says in chapter 1, there doesn't seem to be any such requirement. Is there some further condition that for any language that would be used by a group, that language must have such logical vocabulary? Again, there is no indication from any of the chapters in Articulating Reasons that there is such a condition. I'm not sure why the expresively impoverished langauges drop out of the picture. One guess is that for any inferentially articulated language, it is always possible to enrich it to include conditionals. Then the argument in ch. 4 would apply. However, this seems to leave open the possibility that in adding such vocabulary the singular terms are changed such that they act symmetrically rather than asymmetrically. Were this the case, then new conditional/negation-free conclusions could be drawn, meaning the conditional/negation were not actually conservative over the previous inferences. If the claim from the first chapter, that conditional/negation are conservative over any field (Brandom keeps calling collections of material inferences "fields"; I don't know why, nor do I know why they aren't just sets.) of material inferences is correct, then the result from ch. 4 would be that singular terms must behave symmetrically. Offhand, I don't know what exactly the status of the conservativity claim is. This line of thought might call it into question, or, if it is solidly established, it might go a ways towards explaining why Brandom draws the conclusion that he does.

Posted by Shawn at 11:22 AM 0 comments Links to this post

Labels: inference, inferentialism

## Wednesday, July 04, 2007

### Rearticulating Reasons: Monotonic City

A while ago, I wrote a post on monotonicity. In particular, a large part of it was on what I called semantic monotonicity, that is, if some formula \phi(P) is true, and in the model the set assigned to P is a subset of that assigned to Q, then \phi(Q). This needs to be slightly changed for inferences. If from premise P you can infer Q, and Q is a subset of R, then from P you can infer R. Intuitively the extensions of the predicates/concepts in the premise and conclusion are ordered by the subset relation. This is different from monotonicity in side formulas, which says that if you can derive a conclusion, then you get the conclusion no matter what further premises are added to the argument. Brandom's picture in Articulating Reasons rejects monotonicity in side formulas. This is one of the properties of material inferences. They can be turned from good to bad by adding more premises. Some material inferences are semantically monotonic. I should say some concepts or predicates are, to be a bit more exact. For example, the inference from Madrone is a tabby to Madrone is a cat, and from that to Madrone is a mammal. What role do these sorts of inferences play in the inferentialist picture? They happen to be a subset of commitment-preserving inferences. There is more to the story. Brandom distinguishes three sorts of inferential relations, which I went over in a previous post. These relations are ordered such that incompatibility entailments->commitment preserving->entitlement preserving, but not conversely. Based on what is said about incompatibility entailments, I think the semantically monotonic inferences are also a subset of the incompatibility entailing inferences. Everything incompatible with the conclusions of semantically monotonic inferences will be incompatible with the premises. From what I can tell, this works for single premise inferences. I speculate that multi-premise inferences are where one finds commitment preserving inferences that are not incompatibility entailing.

Posted by Shawn at 12:48 PM 0 comments Links to this post

Labels: inference, inferentialism

## Monday, July 02, 2007

### How many ways can one interpret "understanding analysis"?

One of the things I wanted to do this summer was flesh out some of my math background a bit since I'm interested in logic but didn't do enough math as an undergrad. For example, I've learned a fair amount of discrete math, but comparatively little continuous math. To fill in the holes in my education, I thought I'd work through a few of the yellow UTM books. I'm nearing the end of Understanding Analysis by Abbott. It is an introduction to single variable real analysis, and I must say I rather like it. There are chapters on infinite series, topology of the real line, functional limits and continuity, derivatives, integrals, series of functions, and some odds and ends on metric spaces, Fourier series, and the construction of the reals. The exercises in each section have a range of difficulty and the proofs in the chapters are fairly explicit. The second to last section of each chapter tends to contain some of the more interesting results, leaving large chunks of their proofs to the reader. One thing I like is that each chapter starts and ends with a little bit of historic and philosophical background, for example the relative timeline of development of the notions of continuity and derivative. Not a lot, but they suffice to provide motivation for why mathematicians were interested in developing these specific notions. My biggest gripe with the book is that there are a few (too many really) small typos in various proofs. Most of them are easy to spot, but a couple left me confused for a spell. I think the book labors too long on the introduction of quantifiers, but maybe it is needed when taking this book as a freshman. In any case, I've enjoyed the book and think I've gotten a decent amount out of it. One of Abbott's aims in writing the book is to bolster mathematical intuition (in the non-loaded sense), and, generalizing from a single case, he seems to do alright. I'd recommend it to any philosopher who wants to get a bit of a feel for analysis.

Posted by Shawn at 9:15 PM 2 comments Links to this post

## Sunday, July 01, 2007

### Like a shabby pedagogue

Did Feyerabend read Perry or Mach? Probably the latter, at least. But here's a nice little story from his Three Dialogues on Knowledge.

"B: ... Some years ago I was walking towards a wall and saw a very disreputable individual coming towards me. 'Who is that bum?' I asked myself - and then I discovered that the wall was in fact a mirror and that I had been looking at myself. At once the bum turned into an elegant and intelligent-looking character."

Posted by Shawn at 10:53 PM 0 comments Links to this post