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Towards an epistemological classification of scholarly disciplines - Beware of the Train [entries|archive|friends|userinfo]
pozorvlak

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Towards an epistemological classification of scholarly disciplines [Jun. 12th, 2008|12:23 pm]
pozorvlak
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  • Mathematics is the study of statements which can be proved.
  • Science is the study of statements which can't be proved, but can be falsified.
  • The humanities are the study of statements which can neither be proved nor falsified, but whose credibility can be supported or undermined by advancing evidence.
  • Philosophy is the study of statements which can neither be proved nor falsified, and for which evidence cannot be advanced.
This classification suggests some further ideas. Firstly, mathematics is in some sense the easiest branch of scholarship: in fact, mathematics is precisely "that which is easy", for the appropriate definition of "easy". Secondly, philosophy is really, really hard. This accounts for the almost total lack of progress in philosophy in the last 2,500 years. Philosophers are still debating problems posed by Thales of Miletus, and defending (or attacking) positions advanced by Plato; pretty much all we've achieved is to clarify our statements of the problems1. Is there a single statement whose truth would be agreed-on by all philosophers? I'd love to be corrected, but I don't think there is. Compare the progress achieved in philosophy to the progress achieved in mathematics over the same time period, or with the progress achieved in science in a mere 500 years, and you'll see what I mean. As far as I can see, all progress in philosophy has come by re-stating philosophical questions as scientific or mathematical ones. And this despite philosophy attracting some of the best and brightest minds of every generation. Hell, even the humanities people are arguing about different books now. Thirdly, mathematics is neither a science nor a branch of philosophy, though it has things in common with both.

We're left with a puzzle, though: empirically, mathematics is difficult, when it ought to be easy. I'd like to suggest several reasons for this. Firstly, mathematics is very old, and has been worked on in the past by beings of otherworldly intelligence: all the easy and accessible problems were solved long ago, mostly by Euler. The git. These days, even finding a sufficiently easy problem is challenging for us mere mortals. Secondly, much of mathematics is highly abstract, and humans are not evolved for highly abstract thought: the capacity to grasp concepts with high degrees of abstraction (which is not the same thing as intelligence) seems to be quite a rare one, and requires substantial training to be brought to a useful level. Thirdly, performing experiments in mathematics was largely impractical until the invention of the computer, and even today the technology for performing mathematical experiments is at an early stage of development. This means that until recently our experience was limited to those systems which can be worked out in the head or on paper.

1 This is, of course, a slight exaggeration. For instance, the alert reader will have noticed my implicit appeal in point 2 to Karl Popper's principle of falsifiability: Popper's theories have greater credibility and explanatory power than those of the logical positivists, and thus represent an advance in the philosophy of science. But I bet you could find a philosopher who disagreed with it without too much difficulty, probably just by walking into any philosophy department common room and declaring your support for the principle in a loud voice. Philosophers are an argumentative bunch. For comparison, try finding a mathematician who doesn't agree with Cauchy's residue theorem, or a physicist who doesn't agree that general relativity represents a good approximation to reality.
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[User Picture]From: half_of_monty
2008-06-12 11:56 am (UTC)
I like this. Ta.

One extra comment on the horrors surprising difficulty of mathematics: it's much more cumulative than the others. Okay, this applies also to various parts of the hard sciences, but in the majority of disciplines, you can tell a layperson what question you're trying to answer without too much difficulty. In mathematics, not only can you not tell a layperson, but you can't really tell the advanced mathematician across the office from you either, at least not without going back a long way. This is because every new piece of theory you may wish to learn requires textbooks of previous bits.
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[User Picture]From: pozorvlak
2008-06-12 12:32 pm (UTC)
Oh, yes, this is a big problem, particularly since the explosion of mathematics in the twentieth century. I'm very lucky in that I can at least motivate my problem (though not the techniques I use to attack it!) to laymen without too much handwaving :-)
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[User Picture]From: michiexile
2008-06-12 12:17 pm (UTC)
First off, I agree with half_of_monty above - it's a neat system.

Can you think of any reasonably well-known scholarly discipline that doesn't fit in this schema? I'm struck by how to some extent all of theology is subsumed in Philosophy.
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[User Picture]From: pozorvlak
2008-06-12 12:36 pm (UTC)
Thank you :-) The definition of science is reasonably standard, and the definition of mathematics isn't new (even if it isn't widely agreed-upon), but I think the other two are a fairly natural extension of the scheme.

Treating theology as a branch of philosophy sounds about right - though in Jewish/Christian/Islamic theology you can appeal to the text of holy books, so it has some aspects of a humanities subject. I don't think engineering fits neatly into the scheme. The study of languages is an interesting one: linguistics is a branch of science (or is it, by this definition? I'm not sure), but the study of literature is a branch of the humanities.
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[User Picture]From: neoanjou
2008-06-12 12:49 pm (UTC)
Interesting;

There are suggestions that we should remove the idea that falsification is central in Physics, given that it bears little relation to the way that Science is actually done (which is more a Bayesian accumulation of evidence modifying the preferred hypothesis). And certain ideas, which in principle may be falsified - such as string theory or some cosmologies - have such a broad range of tuneable parameters that it is unlikely that they can be falsified, and will most likely, if they are, come to be rejected by virtue of application of Occam's razor.

Having a Bayesian idea of the process of how science works also provides a neat example of extraordinary claims require extraordinary evidence - if a hypothesis is well supported, even a result which almost falsifies it (i.e. most of the time a falsification won't be of the kinda that seeing a heads falsifies the hypothesis that a coin has two tails, but a result which is 2-5 sigma away from a prediction) should not change it as the dominant idea.
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[User Picture]From: pozorvlak
2008-06-12 01:20 pm (UTC)
Interesting!

The relationship between truth (or rather, credibility) and proof in mathematics is also rather more complicated than I've suggested above :-)
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Falsifiable - (Anonymous) Expand
[User Picture]From: necaris
2008-06-12 12:59 pm (UTC)
Is there a single statement whose truth would be agreed-on by all philosophers?
"We don't know anything", perhaps ;-)? No doubt they would all agree that philosophy is worth studying, at least...
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[User Picture]From: hildabeast
2008-06-12 01:14 pm (UTC)
well first you need to define what you mean by "worth"...
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[User Picture]From: necaris
2008-06-12 01:08 pm (UTC)
Was this post inspired by that webcomic?
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[User Picture]From: totherme
2008-06-13 05:25 am (UTC)
Philosophy is the study of statements which can neither be proved nor falsified, and for which evidence cannot be advanced.

Philosophers are an argumentative bunch

Is academic philosophy actually the study of philosophical topics (with the implication that one's goal is to arrive at an intellectual consensus about them)? Or is it actually (in practice, regardless of what the university prospectus says) the study of tools with which one can argue about philosophical topics?

I sometimes wonder if the goal of computer science is actually to improve the state of technology in the world, or if it's really about improving intellectual with which we can argue religious points. Points like linux vs mac vs windows, or static vs dynamic...
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From: yrlnry
2008-06-18 05:26 pm (UTC)

On the difficulty of mathematics

You suggested some explanations for this, but I think the correct explanation is rather more banal. It's this: mathematics is difficult because everything is difficult. Everything is difficult because we tend to investigate problems that are out at the limits of our investigational abilities. Or to put it another way, mathematics is hard because we're always trying to solve the hardest problems we can solve.

But this is banal because it's not unique to mathematics; in fact it's universal. In engineering disciplines, for example, we're always trying to design systems that are out at the limits of our abilities. Maybe we're trying to design a bridge that is the longest or the biggest bridge ever. That's hard. But say we only need a small bridge. Then we'll try to design the cheapest small bridge we can, and that's hard too.

This is also the essential reason why there are no feasible get-rich-quick schemes: For any value of x, if x were easy, everyone would do it. Consider, for example, the immense competition that now exists in the field of advance fee fraud. Anyone wanting to make a living at this will have to be prepared to work extremely hard.

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[User Picture]From: pozorvlak
2008-06-20 01:45 pm (UTC)

Re: On the difficulty of mathematics

I think I was reaching for that with my first point, but you've expressed it much more clearly than me. So, I only think mathematics is hard because I haven't tried to do anything else to the same level?

It's an interesting thought: is the difficulty of doing original work in field X truly independent of X? Is it as hard to be a really good beautician as it is to be a really good mathematician? How do you even measure that, given people's varying talents?
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[User Picture]From: oedipamaas49
2008-06-28 10:30 am (UTC)
Unrelatedly - I've just added you as a friend, because you write lots of interesting posts. Don't worry, we don't know each other, and I will go away if for some reason you want me to.
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[User Picture]From: asakiyume
2008-11-21 05:22 am (UTC)
I came to your post by most mysterious means, but I must say, I love your initial classifications.
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[User Picture]From: pozorvlak
2008-11-27 07:22 pm (UTC)
Glad to be of service :-)
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