Wednesday, September 19, 2012

A consideration on the subject of math

It seems to me that substance is in some way the subject of every science. This is a topic that I've been concerned with lately, mostly because of claims made about the nature of mathematics. The most common claim is that math is the science of quantity (a reasonable claim) and that further developments in math turn it into a science of relation (a reasonable step, granting the prior claim). Yet, it seems to me, that if math is to remain a science about reality, it must in some way include the notion of substance.

But what substance? The continuum, that is, substance considered only under the aspect of having dimension. Math considers substance with regard to its quantity and those qualities and relations that arise from quantity alone. Thus sensible qualities, which in some way involve the activity of a substance or the capacity to be sensed by animal, do not belong to the consideration of math. Motion is a little more difficult, since it can be considered apart from sensible quality, yet it seems to be unintelligible without an end. Then again, when used in math, and end is often supplied. Is this something artificial? That seems more likely to me. The mathematical objects considered by Euclid or Apollonius do not in any way involve motion (perhaps slightly for postulating certain subject), and therefore it is fair to say that what they are doing does not involve the notion of the good. In physics, on the other hand, motion is always considered. Yet nature does not act in vain, so this motion must always be for the sake of an end and therefore involves a good.

What then of the motion in, say, the calculus? Here it seems that we choose a terminus of the motion ourselves. For example, we look for the limit as the change of x approaches zero. We do this because we want to know a certain value that cannot be found otherwise. Yet this terminus is not natural in any way, and so no motion would be toward it without our considering it.

Perhaps a more common art would make this clearer. There's nothing in the nature of peanut butter, jelly and bread that would naturally bring them into proximity with each other. On the other hand, a man, perceiving the good that would arise from such a union, can take the steps necessary to move them together and bring about an artificial good. When limits are taken in the calculus, the terminus seems to be this kind of good. Thus, calculus (insofar as it involves motion) is an art, for the good is posited by man and not found in the nature of the thing.

It might still be asked if there is anything scientific about the calculus, and I would say there is. The derivative of the integral of a function is the same with that function. This is true and makes no reference to the good. Some define integrals and derivatives by limits, but such are not the definition, rather a way of finding them. (There is a similar case in Euclid's book 5; he defines same ratio by a property which lets one know if ratios are the same, yet it seems rather clear that this definition does not penetrate the nature of same ratio.) And when one speak of a way of finding, one is speaking of art. Now this art seems entirely necessary for coming to conclusions in calculus, yet it does not seem to belong to the science in the fullest. (Just as Ptolemy needs to construct his table of chords and arcs in order to proceed, yet the reason he picks what values he does comes down to his own will, that is, they are not natural values.)

Another question arises from my initial study of book 6 of Aristole's Physics. Here he is making propositions about continuity and the composition of the continuum. Would these propositions belong to mathematics, since they do not seem to rely on the qualities proper to natural bodies? And then there are the propositions about motion. Aristotle seems to think that these belong to physics, yet they do not attend to sensible qualities or the kind of substance moving. If any motion belonged to the study of math, it would seem that these propositions would certainly be called mathematical.

Another possibility is that mathematics takes up the existence and nature of the continuum from physics, and therefore is not concerned with giving demonstrations of it. This seems plausible, since the demonstrations in the Physics (at least as far as I've read) don't have anything to do with the things considered in Euclid, neither about ratios nor about shapes. There is some consideration of the finite and the infinite, but Aristotle made clear earlier that these go with a consideration of motion. It is also the case that physics (natural science generally) is typically considered the science about mobile beings. Thus if math tries to take motion for itself, what is left for physics?

Another question arises from the consideration of the mixed sciences, such as in the work of Ptolemy or Newton. There the subject matter is this or that planet, yet the method bears a great resemblance to that of mathematics. This would take a further consideration.

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