Monday, September 21, 2009

My first hour in a Geometry class

This is basically a longer form of the signature quote at the bottom of the page, originally created by a poster named HRG on CARM. My personal attitude toward mathematics is much closer to some combination of formalism and fictionalism than any other position, and I find it useful to remind students that what we are studying is doesn't have to be a perfect representation of the real world in order to be valuable. Indeed, since at the very least the parallel postulate doesn't hold for reality, we are not studying such a system.

So, I start of talking about three criteria that we would like any sort of knowledge to possess. The differ slightly from the usual description of epistemology.

We prefer knowledge that reflects the world in which we live, one that has an application to things outside of our imagination. Saying that a ball is red, that an action is evil, that 2 + 2 = 4 would ideally say something about the apple, the action, or the numbers themselves.

We prefer knowledge that we can show, by some sort of objective process, holds to be true. This allows us to convince skeptical people that the knowledge is valid. Saying that a ball is red, and action is evil, or that 2 + 2 = 4 would mean we had accepted means of demonstrations so that every observer who accepted such means as valid would accept the result.

We prefer knowledge that won't change with time, that stands firm. when something is true, it should stay true and not become partly true or even false later on. A red ball should always be red (assuming the ball itself does not change), an evil action should always be evil, and 2 + 2 should always equal 4.

The next part discusses what I see as three different systems that focus on knowledge using the desired attributes above. Of course, any truly broad system of inquiry will make us of all three of the systems I discuss below, but generally only one of them at time. Like a three-way combination of oil and vinegar, they don't seem to mix together. In the examples, I try to offer samples where the primary dependence seems to match the system I use, and the other systems are of secondary significance.

Empirical systems
Empirical systems rely on the investigation of the real word and use inductive processes and assumption to decide what is more generally true about that world. They use demonstrable methods such as taking measurements, running controlled experiments, and making predictions to be verified by further observations and experiments. Because the observations are founded in actual investigations, their reality to the world is a natural consequence. However, the use of the inductive method means that the results can not be certain, that all finding are provisional and possibly erroneous. The best that can be hoped for is beyond a good reason to doubt a finding. Examples of empirical systems would include physics and the results of national polls regarding elections.

Formal systems
Formal systems rely on working from a starting point accepted as being true and using an accepted, objective, deductive means of deriving new truths within that system. The method of deriving new truths within the system is the demonstration of that truth, and the process of deriving the truth would be repeatable regardless of the observer. Also, given a specific starting point, the same truth will always be the result (at least, ideally). However, since the starting point is selected rather than derived, there is no guarantee that such a starting point will have any bearing on the world around us. A formal system may be a useful model of reality, but we can never know whether it matches reality. Examples would include philosophy and law (as practiced at the appellate level).

Belief systems
Belief systems use the trusted, revealed knowledge of an authority of some sort to provide as description of the world and its operation. The truths that we get from this authority are accepted because of the level of trust, not because of any independent test that we run. Belief systems do provide a view of reality and knowledge about the state of affairs in the universe. Also, because the source is some consistent authority, they have the feature of certainty, of not changing within the system itself. However, because our acceptance of this knowledge is based upon trust rather than a means to validate them, these truths are generally not demonstrable to others in a significant or meaningful way. Examples include Shintoism and Objectivism.

Then the question that guarantees a minute of silence and ten more of discussion: which one of these is mathematics?


J said...

Useful distinctions, sir. However I feel that formalism of whatever often tends to reinforce anti-humanism. Of course we need mathematics--Im no postmodernist--but the mathematics is not an end in itself. Calculus allows engineers to build bridges, or airplanes, or alas, bombs; it may also help with economic research. The equations do not float in some platonic abode.

So there is a pragmatic aspect to math and science, obviously. We don't have to be William James to understand that. Formalists often (especially logicians and programmers) lose sight of the practical applications--and in some circumstances, it's not really a matter of demonstrable truth, but something like efficacy (ie a new medicine--does it work, or not).

One Brow said...

I absolutely agree with you here, and I emphasize that even while the math itself is a formal system, the primary reason for studying it is the usefulness of the system involved. I'm a formalist, but a very pragmatic version (if that makes any sense to you). So, formal systems don't have any gaurantees concerning reality, but we still aim for it (much like we try to aim for certainty in science and demonstrability in our belief systems).

J said...

Saying that a ball is red, and action is evil, or that 2 + 2 = 4 would mean we had accepted means of demonstrations so that every observer who accepted such means as valid would accept the result.

I find it interesting (and somewhat anomalous) that you consider an ethical statement--"Action x is evil"--a statement of objective truth, similar to arithmetic (2 + 2 = 4). That is a minority position. Carnap and Co claimed ethical statements such as "killing is Evil" were meaningless (really Hume and Darwin had suggested as much).

I don't agree with the Carnapian positivists in regards to the meaninglessness of ethical/normative statements (or with the scoundrel Hume), yet I think the problem with providing a satisfactory definition of normative language remains--and is all the more problematic for secularists, Darwinists and naturalists of whatever sort.

Given naturalist assumptions, any definition of justice seems primarily "hedonic": ie good is pleasure, bad is pain. That might work for certain situations--say, consumerism, taste in sports teams, cars, music, etc--but people obviously do use moral language to denote objective morality, the Good, Justice, etc., even if they can't really prove that objective morality exists.

When some person on KOS starts ranting that "Bush is Evil", he does not mean "I don't care for Bush, but you can form your own opinions", does he? No. He means Bush is Evil according to some standard of justice--not merely shared or agreed upon, ie hedonic. For hedonic ethics (whether utilitarian, or any secular ethics) merely implies you have decided Bush offends you (causes you uneasiness, displeasure--). Things could be different, given a hedonic starting point, due to conditioning, background, nationality, etc. Hedonic ethics IS relativistic ethics, regardless of utilitarians/humanists' attempts to persuade us otherwise.

Ergo, I think ethical universals, like Justice, do exist in some sense--and that hedonic ethics/utilitarianism is mistaken--though proving that may be difficult if not impossible. When we say "Stalin was Evil incarnate" we are not merely barking, or playing some syntax music as the positivists suggested. I am reluctant to bless some full-blown platonism, so I would say it's sort of a psychological universal--perhaps Sanity--rather than metaphysical/transcendent---but I am not entirely convinced a metaphysical account is impossible...

Uri said...

but the mathematics is not an end in itself.

Oh, yes it is. Like any creative field of human endeavour. How horrible it would be if the only justification for things were bridges, airplanes and economic researcg!

Uri said...

Blarg. research. Leave it to me to undermine my rhetoric with a typo.

One Brow said...

Welcome Uri, and thanks for the comment.