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October 15, 2008

Berlin vs. Göttingen

For me, the eye-opener in The Architecture of Modern Mathematics was the suggestion that the 20th disputes of the foundations of maths (platonism vs. intuitionism, etc) were, in part, confusing after-echoes of a pivotal conflict in late 19th German mathematics between the the work done in the universities of Berlin and Göttingen: more specifically, between the different approaches to complex analysis taken by Karl Weierstrass and Bernhard Riemann.

There is a story about the rigorization of the calculus wherein doubts over the status of its concepts where resolved by progressively expelling geometric intuitions in favour of strict arithmeticization: starting with Cauchy, but happening decisively with Weierstrass.

Riemann's work went the other way: finding new forms of geometric thinking with which to understand complex analysis. Weierstrass, for whom geometry didn't even properly belong to pure mathematics, would have none of this. So whilst Riemann's results were in advance of others, the party line in Berlin was that his methods were invalid, and that anything of genuine value that he might produce would have to be re-worked with appropriate rigour.

The conflict can be characterised as between computational and conceptual understandings of mathematics. So in Berlin, functions were taken as power series, and the treatment of potentially problematic mathematical entities (irrationals, imaginary numbers) was to say that if they could be represented canonically and if it was known how to calculate with them then that was all that mattered. Whereas in Göttingen great value was placed on having representation-independent understanding of mathematical entities, and on being able to obtain results by “pure thought” rather than by calculation.

Riemann died relatively young, and there were differences amongst those who regarded themselves as carrying on his work; one can distinguish an intuitive-conceptual (Klein) from an axiomatic-conceptual one (Hilbert). Also, it seems that part of the motivation for early 20th century investigations of the foundations of mathematics was not so much carrying forward the Weierstrassian spirit as attempts by those of the Göttingen party to show that rigour wasn't confined to Berlin.

This three-fold distinction makes much more sense to me than intuitionism/formalism/platonism. (This latter trinity is just a bit odd, intuitionistic logic being no less formal that classical logic.) It might be interesting to see how it could be projected further back: e.g. John Wallis the computationalist, Leibniz the axiomatic-conceptualist and Principia-era Newton the intuitive-conceptualist.

Posted by robin2 at 10:38 PM | Comments (0)

October 14, 2008

Screwdrivers

Years ago I read a book that was the transcript of some seminar, chaired by Umberto Eco, that was about the relationship between readers' interpretations of books and authors' intentions. (There may have been more to it than that: I forget.) In it, Richard Rorty took the view that a reader could interpret a book any way they saw fit, and no-one could rightly say that they'd got it wrong.

Rorty gave the following analogy. A spreadsheet is a piece of software that is intended to be used to do calculations, whereas a word-processor is intended to be used for writing documents. However, if he found that he could write documents perfectly well using a spreadsheet then that would be a perfectly valid thing to do.

Someone countered, saying that although Rorty might not be interested in the difference between a spreadsheet and a word-processor, there was a legitimate field of human enquiry—Computer Science—that was concerned with establishing these distinctions. (So that is what computer scientists do.)

My thought was: “It won't sound so clever when you're explaining it to I.T. support. And I bet you'd use a screwdriver to a open paint can.”

More recently I was listening to a recording of Hubert Dreyfus' introductory lectures on Heidegger. In talking about the nature of equipment, there was (inevitably) a discussion of hammers: a tool is defined by its use, and so a hammer is at its hammer-most when its on the job—being used for hammering—but that this won't show up if it considered as an object with objective properties.

One student wanted to make a point about equipment being used for more than one thing. Presumably unfamiliar with claw-hammers, he gave the example of a screwdriver, which is normally used for driving screws but that someone might use as a lever, when taking the lid of a paint can.

There are a number of questions that might be raised at this point:

However, there is a more fundamental point to be made, that philosophers and their students might want to consider. DON'T OPEN PAINT CANS WITH A SCREWDRIVER. IT BREAKS THE BLADE, AND DECENT SCREWDRIVERS AREN'T CHEAP. USE A PAINT-STIRRER, A 2P COIN, THE HANDLE OF A FORK, OR ANYTHING ELSE. JUST NOT A SCREWDRIVER.

Thank you. I feel better now.

Posted by robin2 at 11:16 PM | Comments (0)

October 13, 2008

Probability and the infinite worlds

In Potentiality and virtuality (Collapse II), Quentin Meillassoux discusses Hume's Problem: there is no logical reason to believe in fixed laws of nature, as any formulation of such laws would be on the basis of empirical observation, and there there is no logical reason to think that the past is a good guide to the future. On this account, there is always the possibility that the ‘laws’ of nature might suddenly change tomorrow.

Meillassoux takes on a counter-argument that if changes in the laws of nature were possible then they surely would have happened. They haven't, so the possibility would have to be regarded as so unlikely as to be non-existent. This can be re-stated so as to make it's circularity obvious: the past can be taken to be a reliable guide to the future because it always has been previously.

Meillassoux takes a different tack. In order to be able to assess how likely it is that the laws of nature could change—given that no such changes have taken place so far—we would have to know the prior probability of the universe having fixed laws or not. (Similarly, if someone wanted to ascribe significance that physical constants are “just right” to allow for the development of life then they'd have to understand the probabilities involved there.)

So one has to consider all the possible universes to know how significant it is that we got this one. For Meillassoux, merely declaring that all this is outside the valid scope of probabilistic reasoning, or that the full range of possible universes is empirically unknowable, would be too weak a result. It would leave Hume's Problem where it was: as a lack of certain knowledge rather than as positive knowledge of a lack of certainty. So instead he contends that it would not be possible to draw conclusions from the universe being as we find it, even with a purely theoretical understanding of the range of possible universes.

At this point I lose the thread of the argument. For Meillassoux, if we claim to imagine an infinity of possible universes then we must acknowledge that—since Cantor—mathematics provides infinities in different sizes. Therefore “since there is no reason […] to choose one infinity rather than another” then probabilistic reasoning is impossible.

I'm obviously missing something fairly vital here. The article expands on the theme of sets and infinity, so this must be key to the point that is being made. But at this juncture the argument comes across like someone trying to crack a walnut with a sledgehammer, and missing. For now I'll just counter with the two points I've got at the moment.

The first isn't to do with the counter-argument to Hume's Problem, but with the Goldilocks argument alluded to above, that Meillassoux treats as being part of the same “probabilistic sophism”. The argument as I understand it is that if the constants of physical laws had been only slightly different from what they are then life, including human life, would have been impossible: and so it strains credulity that they could be as they are by chance. So what is being envisaged here is that the range of possible universes is determined by the ranges of possible values for a finite set of real-valued physical constants. In which case it isn't true that the infinity of possible universes cannot be known: it is that of the continuum.

The second is a bit whimsical. It seems strange to suggest that it might be theoretically possible to know everything about the possible universes apart from the size of infinity of possibilities, but suppose this is granted. Now suppose the possibilities can be ordered into a sequence, so that each possibility can be assigned an ordinal (which may be transfinite). Also, suppose the following:

Assuming that the number of possibilities isn't finite, and that they are all equally likely, then the probability of the universe having fixed laws would surely be 1 in 3, regardless of the exact length of the sequence.

Posted by robin2 at 12:00 PM | Comments (2)

October 12, 2008

Holiday Reading

This year my holiday reading included bits of Collapse covering speculative realism (thanks, Robin), some old papers by Rodney Brookes on Artificial Intelligence and robotics, and The architecture of modern mathematics—Essays in History and Philosophy eds. José Ferreriós and Jeremy Gray. All thought-provoking stuff. Unfortunately I've lost track of the thoughts provoked. I'll see what I can do, though.

Posted by robin2 at 12:00 PM | Comments (0)

October 05, 2008

Biology

Note: I wrote this a while back after having read the interview with Alain Badiou in Collapse I, but didn't post. I don't know quite what I mean by “Bergson, of all people.”

Badiou's claim that biology cannot yet be counted as a science shows that he has a rather idiosyncratic notion what science is. The complaint is that biology is a morass of observations and specific theories: these may give rise to plenty of useful knowledge, but nonetheless biology lacks an over-arching conceptualisation that would give the subject meaningful order. The concepts of biology “fail completely to present the phenomena concerned in the register of eternal truths.”

I don't want to get into the question of whether the requirement for eternal truths is a merely philosophical imposition; nor the question of whether Darwinism might speak in the register Badiou wants to hear (nor whether this would be a good or a bad thing). Instead I'll reminisce for a bit about why I never cared much for biology at school.

There were superficial reasons; such as the variety of dead things in pickling jars displayed on rickety shelving around the lab. But it was also that what we were taught seemed to be without rhyme or reason. Whereas with chemistry almost everything at that time seemed to be tied in with the periodicity of the elements, biology was just one thing after another: pollination, the set of little bones in the ear, the carbon cycle, the circulation of the blood, etc, etc. It sometimes seemed that the main skill required was to be able to pick up on all the vocabulary involved.

In reality the main skill was to be able to correctly label the diagrams that featured in the syllabus: this much as explained to us by one of our biology teachers. She also suggested that it was a good idea to remember the number of labels that each of the set diagrams was supposed to have: that way it would be easier, in exam conditions, to know whether or not you'd missed something out. At the time I think I was rather shocked that a subject could require you to treat its content so cynically. In retrospect it was probably perfectly good advice: if you like biology, and you want to study it further, then it makes sense to use a bit of cunning in order to get through the more elementary material. I couldn't understand how it would happen that someone would decide that they liked biology; to some extent I still can't.

The closest I've ever come to acquiring a taste for this sort of thing was with the smattering of cognitive neuropsychology included in my undergraduate degree. I recall enjoying the difficulty of it: the careful thought needed to see what it was that some theory was predicting or that some evidence was showing. Badiou is even more damning here: “Even more so than biology, [cognitive science] is just a mass of facts and techniques, devoid of concepts or adequate formalisms […] It is no more advanced in its understanding of the phenomena than was Gall's phrenology.”

At university I was particularly struck by the theorising about implicit learning and memory: the idea that it is possible for someone to learn something without being able to give an account of their knowledge, or even being able to remember having learnt. It is apparently possible for amnesiacs to acquire new skills. In 19th century psychology it was thought that the brain must contain memory traces that could influence behaviour but were either too weak to reach consciousness or had become dissociated from the ego. But in the 20th century it became generally accepted that some memory must differ in kind from that normally considered. It seems that this starts with Henri Bergson, of all people: “The past survives under two distinct forms: first, in motor mechanisms; secondly, in independent recollections.” (See Schacter, D. L. (1987). “Implicit memory: history and current status.“ J of Exp Psych: LMC, 13 for more details.)

This, of course, can be of no importance for Badiou: if truth is to be distinguished from mere correct judgement, then it cannot be understood by looking at the mechanisms of knowledge (be they neural or institutional). Nonetheless, it can't be completely without interest. The distinction between two different kinds of memory, as well as being noted by Bergson, shows up in 20th English philosophy as the difference between ‘knowing that’ and ‘knowing how’, and loosely corresponds to ‘present at hand’ vs ‘ready to hand’ in Heidegger. Might this not qualify it as being a decisive event in the history of philosophy? (My knowledge of Bergson and Heidegger is vague and largely second-hand—but my understanding is that they treated this distinction as being much more than some arbitrary psychological fact.)

Posted by robin2 at 02:17 PM | Comments (0)