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Towards an epistemological classification of scholarly disciplines [Jun. 12th, 2008|12:23 pm]
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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)

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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.
[User Picture]From: pozorvlak
2008-06-12 12:32 pm (UTC)

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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 :-)
[User Picture]From: michiexile
2008-06-12 12:17 pm (UTC)

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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.
[User Picture]From: pozorvlak
2008-06-12 12:36 pm (UTC)

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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.
[User Picture]From: necaris
2008-06-12 12:57 pm (UTC)

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Doesn't pretty much all theology have texts to which you can appeal? Apart from the obvious Semitic religions, there are Taoist and Confucian core books, and the Hindu Vedas...
[User Picture]From: pozorvlak
2008-06-12 01:12 pm (UTC)

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Yes, I suppose so. Good point!
[User Picture]From: michiexile
2008-06-12 01:37 pm (UTC)

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As for linguistics - it seems to depend a LOT on which linguistics you talk about. Some of it seems to be humanities, quite a bit of it take on the kind of sociological empiricism that pulls it into the sciences, phonetics/phonology et.c. lodge quite solidly into sciences, and parts of the Chomskian syntax theories actually start resembling mathematics.

All within the above given definitions, of course.

And one of the core critiques to the mathematical Chomskyisms seems to be that it decouples from reality. I recall one story from Stockholm University where a Chomskian was giving a seminar, and was challenged on one particular prognostication it made: "You say that all natural languages do $FOO! But Estonian does !$FOO." "Well, in that case, Estonian can't be a natural language."
Just like mathematics.
From: MichaelP-J [co.uk]
2008-06-12 06:28 pm (UTC)

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Engineering isn't a scholarly discipline, which is why it doesn't fit. It's not about the study of something but the solution of practical problems. I once heard Michael Rosen describe how English language "gets other languages down dark alleys and mugs them for all their good words". Similarly, engineering mugs other disciplines for proven results and doesn't much care about how they were proven.

Oh, and an Oxford theologian friend once remarked that theology isn't the study of God but the study of arguing...
From: MichaelP-J [co.uk]
2008-06-12 06:32 pm (UTC)

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Oh, and I notice that I've sloppily said "proven" when you specifically said that science doesn't prove things. In which case I should be even more empirical:

Engineering mugs other disciplines for results and techniques that appear to work reliably. It may occasionally delve into the discipline to establish exactly how reliable something is, but for most engineers if it works, it's fine.
[User Picture]From: pozorvlak
2008-06-12 11:28 pm (UTC)

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I mentioned the idea to my flatmate (whose degree was in psychology and philosophy), and he broadly agreed, but suggested an extra level: Social sciences are the study of statements which in principle can be falsified, but in practice falsification is too difficult.
[User Picture]From: necaris
2008-06-13 08:20 am (UTC)

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I was going to make a point about falsification in social sciences depending on which tower of assumptions you were leaning over from, but on second thought that sounds like it might be true for all sciences?
[User Picture]From: neoanjou
2008-06-12 12:49 pm (UTC)

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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.
[User Picture]From: pozorvlak
2008-06-12 01:20 pm (UTC)

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Interesting!

The relationship between truth (or rather, credibility) and proof in mathematics is also rather more complicated than I've suggested above :-)
From: (Anonymous)
2008-06-17 10:46 pm (UTC)

Falsifiable

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But don't throw out the baby with the bath water. It seems to me that the job of the sciences these days is to produce falsifiable sentences - but that doesn't mean that science needs to progress through falsification. In other words, the practice of science might follow a Bayesian accumulation of evidence model, but the things that people are accumulating evidence for, or against, are falsifiable hypotheses.
[User Picture]From: necaris
2008-06-12 12:59 pm (UTC)

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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...
[User Picture]From: hildabeast
2008-06-12 01:14 pm (UTC)

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well first you need to define what you mean by "worth"...
[User Picture]From: pozorvlak
2008-06-12 01:19 pm (UTC)

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"Philosophers, for instance, are very concerned with the problems posed by real life, such as 'what do we mean by "real"?' and 'how can we arrive at an empirical definition of "life"?'" (Douglas Adams)
[User Picture]From: hildabeast
2008-06-12 01:23 pm (UTC)

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but those are very interesting questions! consider when people online talk about "real life" meaning "offline life"...

ahhhhh philosophy... i like you more than maths.
[User Picture]From: pozorvlak
2008-06-12 02:07 pm (UTC)

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I never said philosophy wasn't interesting, I just said it was unfeasibly hard :-)
[User Picture]From: shuripentu
2008-06-15 04:45 pm (UTC)

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Nah, Descartes for example wouldn't agree with that. ;p
[User Picture]From: shuripentu
2008-06-15 04:46 pm (UTC)

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Er, by "that" I mean "the statement that we don't know anything", not that "the statement that philosophy is worth studying". Although there are almost certainly philosophers who think the latter too.
[User Picture]From: pozorvlak
2008-06-15 06:42 pm (UTC)

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Wittgenstein springs to mind, though his ideas were of course more subtle than that.
[User Picture]From: necaris
2008-06-12 01:08 pm (UTC)

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Was this post inspired by that webcomic?
[User Picture]From: pozorvlak
2008-06-12 01:13 pm (UTC)

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The idea had been floating around in my head for a while, but that was one of the things that prompted me to write it down :-)
[User Picture]From: totherme
2008-06-13 05:25 am (UTC)

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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...
[User Picture]From: winterkoninkje
2008-06-18 05:40 am (UTC)

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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?

Of the philosophers I've met, they'd agree with that (barring a few of the more argumentative ones who'd lapse into metadialogue, thus proving the point). At the same time, philosophy should be distinguished from political science, law, and rhetoric which also teach the skills of argumentation.
From: yrlnry
2008-06-18 05:26 pm (UTC)

On the difficulty of mathematics

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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.

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

Re: On the difficulty of mathematics

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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?
From: yrlnry
2008-06-23 01:38 pm (UTC)

Re: On the difficulty of mathematics

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Here I think you've opened at least one orthogonal issue. The work of a beautician is essentially different from that of a mathematician because the work of a beautician has a component of moment-to-moment performance that is absent from mathematics. It has this in common with craft activities (like, say, cooking or shoemaking), performance arts, and sport. The outcome of the beautician's work depends not only on design, advance preparation, training, and so on, but on the specific execution of that particular piece of work at that particular instant. You could be the best beautician in the history of the world, but if you are having a bad day, you might do an uncharacteristically bad job.

Also, the beautician's work product is temporary while the mathematician's is permanent. We have legends of the acting of the divine Sarah Bernhardt, but we don't know what her performances were actually like. Maybe Julia Roberts is better; who can say differently? The beautician's work is similarly ephemeral. But Euler's best work is still with us, a permanent yardstick against which you can compare yourself and come up short.

[User Picture]From: oedipamaas49
2008-06-28 10:30 am (UTC)

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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.
[User Picture]From: asakiyume
2008-11-21 05:22 am (UTC)

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I came to your post by most mysterious means, but I must say, I love your initial classifications.
[User Picture]From: pozorvlak
2008-11-27 07:22 pm (UTC)

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Glad to be of service :-)