Q&A: The Applicability of Mathematics
The Applicability of Mathematics
Question
In a parallel discussion about the possibility of a finite three-dimensional body that has no center, you say the following:
“There is no difference at all between the questions. Anything that is mathematically possible can also be realized in practice. The question of whether it is realized or not is an empirical question (you have to check and see whether such a thing exists or not).”
This claim seems problematic to me in two respects: on the general principled level and on the specific level (where I’ll bring an example).
On the general principled level, you are basically insisting that there is no purely mathematical domain distinct from the empirical domain (“There is no difference at all between the questions”), but immediately afterward you qualify this by saying that this is conditional on its not conflicting with an existing law of nature and on its receiving empirical observational support. In my opinion, the very introduction of these qualifications contradicts the first part of the sentence, that there are not two questions (=domains).
I also don’t think this criticism of mine is semantic hair-splitting, since you phrased yourself in a very clear-cut and emphatic way.
I’ll present the specific problem in your position through an example from the realm of negative numbers. According to your view, we should have to say that there is no principled obstacle to the existence of a physical parallel to the expression minus three, for example “minus 3 oranges.” As long as this expression does not contradict a law of nature (and it doesn’t), you would have to—again, according to your view—leave this question open. That seems absurd to me, because I see no way at all to actually count (by direct pointing) those “negative oranges,” and certainly don’t see how one could provide a physical description of their properties. So here you have a perfectly valid mathematical expression whose existence—even hypothetical existence—we would have to categorically deny in the physical world. In short: these really are two separate questions.
Answer
As I wrote, mathematics allows bodies to stand in midair, but there is a law of nature that forbids this. So it is indeed possible even in our world, but empirically it turns out that this is not actually realized. I don’t understand what is difficult or contradictory here.
You are talking about a mathematical concept, and I am talking about a mathematical structure or relation. Any spatial structure that mathematics allows can be realized. By the way, quite apart from the above, even minus 3 oranges can be realized, and is realized (when I owe my friend 3 oranges).
In short, there are not two separate questions.
Discussion on Answer
I have a hard time with these conflations.
I never said anywhere that mathematics and physics are the same domain or that they deal with the same questions. On the contrary, I wrote in several places that this is not so. For example, mathematics cannot be refuted, whereas physics can. Physics is at most a model of mathematics.
When I wrote that it is the same question, I was not identifying the domains. The question whether the mathematical structure is coherent and the question whether it can be realized in the world are the same question. What is realized and what is not is an empirical question that has nothing to do with mathematics. But what can be realized is a hypothetical question, and in my view it is the same question as whether it is coherent in mathematics.
My claim was that in principle one can find in physics a model for every coherent mathematical structure. For example, if a finite three-dimensional body without a center is a coherent structure, then there is no obstacle to there being such an object in the world (unless the laws of nature forbid it, and then that is an empirical qualification). That does not mean that every concept defined in mathematics will appear in physics. A model for it may appear, as with debt. That does not mean that there is a concept of debt in mathematics. Mathematics also has no concept of force, but forces operate according to vector calculus. Unlike force, which is a concept (physical), and the number minus 3, which is also a concept (mathematical), vector calculus, for example, is a mathematical structure (not a mathematical concept), and it can appear in physics (and indeed does).
Okay, it seems to me that now you are phrasing yourself a bit differently and have stated things more precisely. What you are saying now I accept.
Still, one important central point is not clear to me: do you think, as I do, that there are structures and/or concepts in mathematics that are a priori prevented from appearing in the physical world? Take again, for example, the case of a finite three-dimensional body that has no center. Suppose the concept of such a body is mathematically valid. Then someone comes along and argues that the very concept of “center” has metaphysical significance that is imposed on the world. Suppose you become convinced that he is right—would you then agree to say that finite three-dimensional bodies necessarily have a center, and therefore even if mathematics “allows” situations of a body that has no center, this is not relevant to physics? That is, the center is a fact that is necessarily present in physics as well.
Logically, yes. Theoretically, if there were a law of nature that did not allow a body without a center to exist (for example, the body would collapse into its center and disappear from the world), then there would be no such bodies. But there is no such law.
I don’t understand how you distinguish between a “mathematical structure” and a “mathematical concept,” but in any case that doesn’t seem to me to be a relevant distinction for our discussion. You presented a sweeping claim according to which mathematics and physics are the same domain (“It’s the same question”), from which it follows that every mathematical “entity” or operation is at the same time also a physical entity or operation. That is simply a straightforward logical conclusion from your own words. So it is not clear how, from such a point of view, one can even speak about empirical phenomena, data, or even inductive inferences (without which there is no room at all for modern science).
But your claim regarding negative numbers especially strikes me as strange. I argued that minus 3 oranges is a mathematical expression (or concept or structure—call them whatever you like) that has no physical counterpart, but I did not argue that it has no use in the practical world, as in the debt example you gave. The absence of such a counterpart proves to my mind that mathematics and physics are sharply distinct from one another (they are not “the same question”).
Are you claiming that the expression “debt” has purely physical meaning? Does debt, for example, have density, temperature, and so on? If your answer is no, then you have joined the camp that draws a sweeping distinction between physics and mathematics, and that no longer fits with your earlier claim that “it’s the same question.”