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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.
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.
June 12 2008, 11:56:28 UTC 3 years ago
One extra comment on the
horrorssurprising 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.June 12 2008, 12:32:01 UTC 3 years ago
June 12 2008, 12:17:56 UTC 3 years ago
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.
June 12 2008, 12:36:53 UTC 3 years ago
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.
June 12 2008, 12:57:32 UTC 3 years ago
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June 12 2008, 12:49:10 UTC 3 years ago
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.
June 12 2008, 13:20:39 UTC 3 years ago
The relationship between truth (or rather, credibility) and proof in mathematics is also rather more complicated than I've suggested above :-)
Anonymous
June 17 2008, 22:46:24 UTC 3 years ago
Falsifiable
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.June 12 2008, 12:59:17 UTC 3 years ago
"We don't know anything", perhaps ;-)? No doubt they would all agree that philosophy is worth studying, at least...
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June 13 2008, 05:25:20 UTC 3 years ago
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 points. Points like , or ...
June 18 2008, 05:40:10 UTC 3 years ago
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.
June 18 2008, 17:26:57 UTC 3 years ago
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.
June 20 2008, 13:45:36 UTC 3 years ago
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?
June 23 2008, 13:38:32 UTC 3 years ago
Re: On the difficulty of mathematics
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.
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