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I've posted some slides of two proof theory talks I've given here to the publications section.

More excitingly, a paper of which I'm the lead author has been accepted to Journal of Biomedical Informatics.  The information for that paper can also be found on the publications page.  I should be receiving proofs in a week; I will post the actual paper once it has been published.

I've also just found out that I won a scholarship to attend the North American Summer School on Logic, Language, and Information in Austin, TX this June.  That looks like a great event!

Related posts:

1. Stanford (and other updates)
2. Summer 2010, CMU Logic and Formal Epistemology
3. Two Talks in Three Days

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While I don't want to jump in to that debate here, I do want to talk briefly about a kind of argument that seemed to come to a few people in the seminar.  It's a kind of Kantian (I dare say transcendental) argument for the existence of at least some broad content.  The argument roughly goes like:

1. Disagreeing presupposes public contents that the disagreement is about.
2. We do disagree.
3. Therefore, some content is public.

Broadness follows because narrow content cannot be public.

One thing worth noting about this argument is that one cannot object to (2).  I would argue that "I disagree that there are disagreements" is in effect an instance of the liar paradox.

Although we were talking about disagreement in the seminar, the argument obviously generalizes:

1. Communication presupposes public contents that are being communicated.
2. We do communicate.
3. Therefore, some content is public.

To put (1) in more Kantian terms: Public contents are a necessary condition for the possibility of communication.

The real point that can be argued is (1).  I think it might be worthwhile to replace "communication" with "meaningful communication", but given that meaning is part of what's at stake here, that's a bit dicey.  It seemed like some students in the seminar wanted to push this line of thought by replacing "communication" with "philosophy" which provides a neat twist in that what we are doing while making this argument is philosophy.  That would make (2) equally irrefutable.  But it might put (1) in even more questionable territory.

In any event, I just wanted to put the argument out there to see what people make of it.  I'm personally unsure where I stand on the wide/narrow debate since I think some arguments from both sides are fairly convincing.  Of course, I've yet to see either side attempt to argue that the other notion of content in general is impossible.  Usually (both Segal and Sawyer do this in their entries in the Contemporary Debates in Philosophy of Mind anthology) one argues that the best current options for the kind of content at stake fail.  But that does not mean no future option will succeed.  This warning seems somewhat analogous to Kyle Stanford's argument that unconceived alternatives pose the biggest threat to scientific realism.

Related posts:

1. How Naturalistic Demarcation Relates to Reflective Equilibrium
2. President and Editor-in-Chief: Prometheus

Below you will find the code of the Haskell implementation.  I will follow the code with some comments about it, including the features of Haskell that this project helped me uncover and that I like very much.  Note, however, that this post is NOT a general introduction to Haskell or a general comparison of Haskell with any other language.  Links to such resources will be provided at the end of the post. First, note that the Haskell syntax for type declarations is e::t instead of OCaml's e:t. (In Haskell, ':' is the list cons operation.)  I have also set up a git repository on GitHub containing all of the implementations.  The Haskell one can be found there as well.

Although I mentioned that 'e::t' are typing declarations, you may have noticed the notation 'beta::Integral a => [a] -> a -> a'. In this case, 'a' is a type variable. Thus the expression 'beta' has a polymorphic type, similar to 'a in OCaml.  Now, everything before the '=>' is a type constraint, that is, constraints that the type variables must satisfy.  There can be multiple such constraints even though all the examples in this case use only one.

What does the constraint 'Integral a' mean though?  Integral is a type class in Haskell. A type class is akin to a Java interface in that it provides a list of functions (and their types) that must be applicable to a type in order for it to be an instance of the class. The Integral type class provides methods for integer division, modulus, and the like.  It inherits from the Num type class (Integral is actually a subclass of Real, not of Num, but that's irrelevant for our purposes) methods for addition, subtraction, and multiplication. Now, particular types are instances of a type class. The only types that are instances of Integral are Int and Integer. Int is akin to standard integer classes, while Integer (as mentioned at the outset) allows for arbitrary precision integers. For more on Haskell numbers, see this page.

Thus, in the case of 'beta::Integral a => [a] -> a-> a', this type says that beta is of type [a] -> a -> a ([a] is akin to 'a list in OCaml) subject to the constraint that 'a' must be an instance of the type class Integral (i.e. must be an Int or Integer). The beautiful thing is that integer constants in Haskell can be either Int or Integer; it will decide when it needs to use arbitrary precision and when not.  (Note: I do not know nearly enough about the inner workings of Haskell to know how it does so.)  So, in the file above, the factorial function can be called both on small numbers like 3 and 5, and as 'factorial 3813474071308470184' on numbers that are outside the range of Int.

A few other oddities that should be pointed out:

• The `mod` and `div` refer to the modulus and division functions of the Integral type class. Surrounding any Haskell function by backticks allows it to be used in infix notation. So 'mod 2 3' is equivalent to '2 `mod` 3' although the latter is more natural to read.
• The case pattern matching at the very end is necessary because the find function in Data.List has a return type of 'Maybe a'.  Thus, if the function finds no element satisfying the given predicate, it will come back with 'Nothing'.  Note that in such a case, OCaml would raise an exception.
• 'listeq' is a helper function that tests two lists for equality.  It also requires that the lists have the same elements in the same order; this requirement is actually critical for the β function to work as intended.
• '\x -> f' is Haskell's notation for lambda abstraction, i.e. for anonymous function declaration.  While I could have inlined the functions 'remlist' and 'beta_godel', I left them separately in order to resemble my previous implementations.  I left them separately in those implementations to remain similar to Enderton's exposition.
• '[0 ... n]' is a very nice way to denote the list of numbers from 0 to n.  Although I didn't use it in this implementation, Haskell has very nice list comprehensions.

To do

Although this implementation does work (I will post examples shortly), more remains to be done.

1. I still need to investigate . Two things directly affect the size of this number: and .
2. Haskell has a type inference engine so that the type declarations in this file are not necessary (although I do find them useful).  I'm curious to see whether Haskell will infer more general types than I've given the functions.
3. I need to add more detailed comments to the source files and clean up the git repository.
4. In due time, I will do real speed testing of the three implementations.

Resources

I refer the reader to my old post, linked at the beginning, for references about the logical details.

• Learn You a Haskell for Great Good! This is a great introductory book that can be read for free online.
• Haskell Wikibook
• Some notes on OCaml vs. Haskell.

Related posts:

1. Gödel’s Beta-function in OCaml and Python

Related posts:

1. Lambda Calculi

In a comment, John Sidles takes an essay from the preface to General Victor Krulak's book First to Flight and replaces every occurrence of the word "US Marine" with "mathematician".  The result is breathtaking and somewhat awe-inspiring:

Why does the world need mathematicians?

Mathematics exists today—flourishes today—not because of what we know we are, or what we know we can do, but because of what the world believes we are and believes we can do.

Essentially, because of the unblemished achievements of mathematics over centuries, the world believes three things about mathematicians.

First, they believe that when trouble comes to the world, there will be mathematicians—somewhere—who through hard work have made themselves ready to do something about it, and do it at once. They picture mathematicians as mature individuals—dedicated members of a serious professional community.

Second, they believe that when mathematicians bend their minds to a task, they invariably turn in a performance that is dramatically and decisively successful—not most of the time, but always. The world’s faith and convictions in this regard are almost mystical. The mere association of the word “mathematics” to a challenge is an automatic source of encouragement and confidence everywhere.

The third thing that they believe is that training in mathematics is downright good for young people; that mathematicians are the masters of an unfailing alchemy that helps convert unoriented youths into proud, self-reliant stable citizens—citizens into whose hands the planet’s affairs may safely be entrusted.

The people believe these three things. They believe them deeply and honestly, so much that they are willing to pay for mathematicians to solve problems, and to teach young people.

Therefore, for reasons that completely transcend cold logic, the world wants mathematicians. These reasons are strong, they are honest, they are deep-rooted, and they are above question or criticism. So long as they exist—so long as people are convinced that mathematicians can really do the three things I have mentioned—we are going to have a mathematical profession.

And likewise, should people ever lose that conviction—as the result of the mathematics community’s failure to meet their high—almost spiritual—standards, the profession of mathematics will swiftly disappear.

Is there a chance that such a thing might happen? I think there is. I think that we ourselves can shake these convictions and the accompanying faith which really sustain us. By a lack of attention we can lose the inspirational personal relation that is shared between our senior members and our rank-and-file. Also, by carelessness or inordinate attention to less important things, we can lose the attributes of professional dedication and unfailing preparedness which, in centuries past, has deservedly made mathematics one of humanity’s treasures.

How serious it is, I don’t profess to estimate to you, but it certainly worries me. It does, because if the world wanted to try, she could get along without a profession of mathematics.

In this post, I primarily wanted to reproduce and share the above piece.  That being said, I will conclude with a few brief comments.  It seems to me that while all three things are generally believed about mathematicians, the second seems to be the most true.  Ever since the mathematization of the sciences in the time from Galileo to Newton, there seems to be a push to mathematize anything and everything. This push, I believe, rests on a belief that by mathematically modeling a phenomenon we imbue said phenomenon with the certainty of mathematical truth.

I do also wonder what other professions could be substituted into the passage for "US Marine".  Of course, given the career path that I will soon be embarking on (I am starting a Ph.D. in Philosophy at Stanford this Fall), I am most curious about whether "philosopher" fits.  Judging by my interactions with a variety of people, their reactions lead me to believe that none of the three beliefs listed above are held in any kind of generality by "the people".  Philosophers are not always associated with hard work, "dramatic and decisive" success is hard to define let alone achieve in philosophy, and not many people feel an urgent need for a philosophical education.  While I would disagree with these assessments (sans maybe the second one), I would like to point out one thing about hard work in philosophy: the same attributes that make one a successful philosopher in modern academia are the same attributes that make one successful in any industry.  Hard work---honest toil for those keen on the phrase---cannot be replaced.  Whether or not that's a good thing I leave for another time.

Related posts:

1. Sponsored, Gateway Disc Sports
2. A Metametaphilosophical Remark
3. Two Ways I See It

In other news, I have mostly completed the research project for which I was awarded a PURA grant.  A poster on the project, entitled Ontological Labels for Automated Diagnosis of Left Ventricular Remodeling, was presented on April 12. See my publications page to download the poster.  In theory, this work will now turn into a peer-reviewed publication.

Related posts:

1. Provost Undergraduate Research Award
2. Ontological Labels for Automated Diagnosis of Left Ventricular Remodeling
3. Announcement: A New Approach to Biomedical Ontologies

In a paper forthcoming in Noûs, Otavio Bueno and Mark Colyvan outline and defend an extension of mapping accounts of the applicability of mathematics which they call the inferential conception. Despite the extensions made to mapping accounts, the inferential conception still agrees with mapping accounts on the necessity of starting from an "assumed structure" of the world. In Bueno and Colyvan's view, since any dissection of the world into objects and relations will be theory-laden (and such dissections are necessary for any form of correspondence between empirical and mathematical structures to exist), "there is no avoiding such an assumption." (They do, however, maintain that there may be a "natural candidate pre-theoretic structure", as in the case of a street map. More on this later.)  In this post, I will briefly sketch an argument that such an assumption can be avoided using results from the burgeoning subdiscipline of the cognitive science of mathematics.  In particular, by properly understanding the genesis of mathematical concepts from embodied experience and everyday cognitive capacities, I will argue that mathematical structures should be understood as being grounded in empirical ones. From this perspective, the applicability of mathematics is an inevitable feature of mathematics and the existence of it as a problem is an historical artifact.

Related posts:

1. Limiting Idealizations in the Inferential Conception of the Applicability of Mathematics
2. Motivating an Historical Account of the Applicability of Mathematics
3. Two Talks in Three Days

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