Listening to Stanley Turrentine’s Easy Walker album (Review here.). Wonderful, bluesy, jazzy, groovy album. Excellent drum work by Mickey Roker and nice swinging piano work by the redoubtable McCoy Tyner. Quite struck.

Didn’t know he started his career playing with Ray Charles, Max Roach and Lowell Fulson.

Like Grant Green, Turrentine is an almost forgotten jazz musician who is just being “rediscovered”. Here’s an interesting anecdote (I love photos and stories about jazz and blues musicians):

Turrentine later told the story of how his father helped him develop his characteristic richly focused sound on his instrument (he actually began on cello, but he switched to tenor saxophone at the age of 11 after he was taken to hear Coleman Hawkins). His father made him stand facing a wall while playing a single note for hours, concentrating on producing the full depth and richness of sound from the horn. The exercise seemed strange and even pointless to the boy at the time, and it was only in later years that he really understood its purpose, and made full use of the foundation which it had provided.

Mosiac Records link on Turrentine.

Link to NPR programme on Turrentine.

Ran for about 20 minutes yesterday evening. 3 rounds round the Padang.

Following my last post on Wang’s Paradox, so, what the hell is strict finitism? I’ll post what little I know of it here.

It’s a philosophy of mathematics normally contrasted with constructivism.

Let’s see whether Dummett’s explanation of constructivism and strict finitism makes any sense:

“Constructivist philosophies of mathematics insist that the meanings of all terms, including logical constants, appearing in mathematical statements must be given in relation to constructions which we are capable of effecting, and of our capacity to recognise such constructions as providing proofs of those statements; and, further, that the principels of reasoning which, in assessing the cogency of such proofs we acknowledge as valid must be justifiable in terms of the meanings of the logical constants and of other expressions as so given.

...Traditional constructivism has allowed that the mathematical constructions by reference to which the meanings of mathematical terms are to be given may be ones which we are capable of effecting only in principle [emphasis mine]. It makes no difference if they are too complex or, simply, too lengthy for any human being, or even the whole human race in collaboration, to effect in practice. Strict finitism rejects this concession to traditional views, and insists, rather, that the meanings of our terms must be given by reference to constructions which we can in practice carry out, and to criteria of correct proof on whic we can in practice prepared to rely…

...Strict finitism was first suggested as a conceivable position in the philosophy of mathematics by Bernays in his article “On platonism in mathematics” (1935).”

More information on strict finitism here, which provides this piece of information crucial to understanding Dummett’s article: Strict finitism rejects mathematical induction. Also check out a post on FOM here, I assume ultrafinitism and strict finitism are the same.

Burning question: Is contructivism the same as finitism?

I had a damn lousy day! Took like 5 hours to draft a memo at work cos my laptop kept crashing! Lost my nicely woven arguments and turns of phrases. What a waste of time! Could have been put to much better use, even if I wasted it away lazing. Argh!

Then I forgot to take my helmet to work! Had to wear a probably unwashed helmet worn by countless other strangers. The sweat smell accumulated by various people in the helmet made me wanna puke! Still, that wasn’t as bad as one of Sovan’s helmet I wore once. At least the driving centre suns the helmets but that particular helmet of Sovan’s probably never saw the sunlight! That was an unforgettably nasty experience, years ago but still unforgettable, the smell I swear, still reeks in my mental nose, I mean my mind’s nose.

And I failed my 8th practical lesson again! Last round was 12 points with one immediate failure when my bike fell off the plank. This round was 42 points with no immediate failure! Talk about being demoralised!!

Bitch! Fuck! $^&+%#&) What a lousy day!

Warning: This post will not make sense if you do not have the original article Wang’s Paradox by Michael Dummett. You also need to know what is a valid argument (an argument whose conclusion cannot be false if the premises are true) and what is a sound argument (a valid argument with true premises).

After some preliminary comments on strict finitism as a philosophy of mathematics, Dummett introduces Wang’s Paradox:

Premise 1:0 is small;
Premise 2 (induction step):If n is small, n+1 is small:
Conclusion:Therefore, every number is small.

This is a paradox because the premises are true, the argument is valid but the conclusion patently false since there are numbers that are not small. According to Dummett, “The paradox is evidently due to the vagueness of the predicate “small”“. But how exactly? Someone who wants to reject the conclusion of this argument has two options, to either show the argument is not valid or one of the premises is not true.

ARGUMENT IS INVALID

Suppose the argument is invalid. Consider these two explanations.

First explanation: the invalidity comes from wrongly using universal generalisation on vague predicates. In other words, we cannot use universal generalisation on premise 2 (which contains the vague predicate “small”) to arrive at the conclusion that all numbers are small.

Response to first explanation: Even so, we still have a problem since we can arrive at the conclusion that k is small, for some k is not small. Furthermore, to declare that universal generalisation is invalid in this context is to violate the meaning of “all”.

Second explanation: the invalidity comes from wrongly applying induction to vague predicates, i.e. inductive arguments are only valid when applied to arguments containing non-vague predicates.

Response to second explanation: Even so, we still have a problem since we can arrive at the conclusion that k is small, for some k that is not small. Just use a series of conditionals and modus ponens [see footnote 1 below]. If we go with the second explanation, we would also have to reject modus ponens as a valid way of arguing and surely, we don’t want to do that. [Comment: Actually, one option is to say that modus ponens is invalid when applied to vague predicates.]

Ok, so both explanations above are unsatisfactory. How about this third explanation? Third explanation: Deny that “in the presence of vague predicates, an argument each step of which is valid is necessarily valid”.

Response to third explanation: This violates our whole concept of what a constitutes a valid argument!

Counter-response: But hang on, a strict finitist (this is where Dummett’s preliminary comments on strict finitism comes in useful) would accept this explanation since for the strict finitist, “a proof is valid just in case it can in practice be recognised by us as valid; and, when it exceeds a certain length and complexity, that capacity fails”.

Counter-counter response: But Wang’s Paradox can also be a paradox for the strict finitist since “it will always be possible so to interpret “small” that we can find a number which is not small for which there apparently exists a proof, in the strict finitist’s sense of “proof”, that it is small, a proof not expressly appealing to induction”. For the full argument, see footnote 2 below.

This concludes my first post on Wang’s Paradox.

Footnote 1: First find some particular n that is not small, then apply premise 2 as often as you need with actual numbers to conclude that that the particular n is small, (i.e. start with these premises: 0 is small; If 0 is small, then 1 is small; to conclude that 1 is small. And then use “1 is small” and “If 1 is small, then 2 is small” to conclude that 2 is small etc…). The upshot is that we can still use modus ponens to generate a paradox for some particular n that is not small. It would seem like madness to reject modus ponens.

Footnote 2: My simplified version of Dummett’s argument follows: Call n an apodictic number if it is possible for a proof (in the strict finitist’s sense of “proof”) to contain as many as n steps. First premise: 1 is apodictic. Second premise: If m is apodictic, then so is m+1. Conclusion: All numbers are apodictic.

Ok, here goes my first substantial contribution to the blogosphere: an exposition with commentary on Michael Dummett’s Wang’s Paradox. [Aside: G.H. Hardy, the famous mathematician once observed in A Mathematician’s Apology that “Exposition, criticism, appreciation, is work for second-rate minds.]

Why this particular article?

First and foremost, to bring clarity to my mind on the structure, content and arguments in the article.

Second, to orientate those who are reading Dummett’s paper for the first time and are struggling with it, especially since something like this is presently not available online and this is not a particularly easy paper to read, like most of Dummett’s philosophical writings.

Third, this paper is widely cited in discussions of the sorites paradox and had a significant influence on shaping subsequent literature and thinking on the sorites paradox.

The subsequent posts on this topic will be very rough drafts and frequently revised.

Met Mark tonight. It must have been a year (at least) since I last saw him. I last saw him before he moved to JB. He gave me a problem on probability related to the Gambler’s fallacy, a problem that Sovan is working on for his Phd. I talked about the problem with Mark, without any great contribution to the problem. But it was nice to talk philosophy for a while.

Would also like to record 3 compliments I received and absorbed yesterday. One from my boss, one from Beez, who said I did not need lessons but only needed to play the guitar more since I’m talented, have the feel and can hold the guitar correctly, one from someone (Beez’s friend) who probably didn’t know what he was talking about when he said I sounded like Joe Pass when I played a walking bass jazz rhythm piece I learnt from a book. He also said I should form a band. Of course I was skeptical of all of them, but their comments only reinforced my assessment that my strength on the guitar is feel but my weakness is skill, which was what I told Beez. All lies of course, but lies that made me happy. So there. Haha. But I’ll probably form a band five years from now once I expand reportoire, improve my technique and improvisation skills.

Beez’s friend felt demoralised about being stuck in a rut. And in an atypically optimistic moment, I encouraged him by saying that one day he will achieve his goal of being a good guitar player and that he should start now because one day working on that goal is one day closer to achieving it and one day feeling demoralised and not working towards that goal is one day further from it.

But I’m so glad Beez set up my guitar properly.

Random update: Am currently transcribing Grant Green’s No. 1 Green Street.

Yay! Managed to run 2.5 rounds (took about 17 min at a slow pace) around the Padang after work despite sleeping at 5 am this morning!

Must read this so that I can learn how to set up Chloe. And the manual too.

And oh yes, I ran last Fri. Managed to run up 3 other colleagues to run around the Padang three times. Must do it again with my new Saucony shoes I got today from the Great Singapore Sale at Plaza Singapura. I’m a convert. Saucony shoes are awesome, it feels just right for my feet. Much better than the Asics I have.