Mathematics applications
In a parallel discussion about the feasibility of a finite three-dimensional body that has no center, you say this:
“There is no difference 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 if there is such a thing or not).”
This claim is problematic in my opinion in two ways: on a general principled level and on a specific level (where I will give an example).
On a principled level, you actually insist that there is no pure mathematical field that is distinct from the empirical field (“there is no difference between the questions”), but then you immediately qualify this when you state that it is conditional on it not conflicting with an existing natural law and that it must receive empirical, observational backing. In my opinion, the very fact that you introduce these qualifications contradicts the first part of the sentence, that there are not two questions (=fields) here.
I also don’t think that this criticism on my part is semantic gibberish, since it was formulated very unambiguously and decisively.
I will present the specific problem in your position using an example from the domain of minus. According to you, we would have to say that there is no fundamental obstacle to the existence of a physical equivalent to the expression minus three (for example, “minus 3 oranges”). As long as this expression does not contradict a natural law (and it does not), you will have to – again, according to you – leave this question open. To me, this seems absurd because I see no way to actually count (by direct vote) those “minus oranges” and I certainly do not see how it is possible to provide a physical description of their properties. After all, you have a kosher mathematical expression whose even hypothetical existence in the physical world we would have to categorically rule out. In short: yes, there are two separate questions here.
As I wrote, mathematics allows bodies to stand in the air, but there is a law of nature that would prohibit this. Therefore, it is possible in our world, but empirically it turns out that this does not happen in practice. I did not understand what was difficult or hidden here.
You are talking about a mathematical concept and I am talking about a mathematical structure or relationship. Any spatial structure that mathematics allows can be realized. By the way, regardless of the above, minus 3 oranges can also be realized and is realized (when I owe my friends 3 oranges).
In short, there are not two separate questions here.
I don't understand how you distinguish between a “mathematical structure” and a ”mathematical concept” but in any case, this does not seem to me to be a relevant distinction for our discussion. You presented a sweeping claim that mathematics and physics are the same field (“it's the same question”) hence every “entity” or mathematical operation is at the same time also a “entity” or physical operation. This is just a simple logical conclusion from your own words. Therefore, it is not clear how, from such a point of view, it is even possible to talk about empirical phenomena, about data, or even about inductive inferences (without which there is no place for modern science).
But I find your claim regarding the field of minus particularly strange. I argued that minus 3 oranges is a mathematical expression (or concept or structure, call them whatever you want) that has no physical equivalent, but I did not argue that they have no use in the real world, as in the debt example you gave. The lack of such an equivalent proves in my opinion that mathematics and physics are radically different from each other (they are not “the same question”).
Are you arguing that there is a purely physical meaning to the expression “debt”? Does debt have, for example, density, temperature, etc.? If your answer is negative, then you have joined the camp that makes a sweeping distinction between physics and mathematics, and this no longer aligns with your earlier claim that ”it is the same question”.
I have a hard time with these confusions.
I never said anywhere that mathematics and physics are the same field or that they deal with the same questions. On the contrary, I wrote in several places that this is not true. For example, mathematics cannot be refuted and physics can. Physics is at most a model of mathematics.
When I wrote that this was the same question, I did not identify the fields. The question of whether the mathematical structure is coherent and the question of 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 opinion, is the same question as whether it is coherent in mathematics.
My argument was that in principle, a model can be found in physics for any coherent mathematical structure. For example, if a finite three-dimensional body without a center is a coherent structure, then there is no reason why there should be such an object in the world (unless the laws of nature prohibit it, in which case this is an empirical reservation). This does not mean that every concept defined in mathematics will appear in physics. A model for it can appear, like debt. This does not mean that there is a concept of debt in mathematics. There is also no concept of force in mathematics, but forces act according to vector calculus. Unlike force, which is a (physical) concept, and the number minus 3, which is also a (mathematical) concept, vector calculus, for example, is a mathematical structure (and not a mathematical concept), and it can appear in physics (and does).
Okay, it seems to me that you are now wording things a little differently and have put things in their proper perspective. I accept what you are saying now.
And I still don't understand an important central point: Do you think, like me, that there are structures and/or concepts in mathematics that a priori prevent them from appearing in the physical world? Take, for example, the example of a finite three-dimensional body that has no center. Let's say that the concept of such a body is mathematically valid. Then someone will come along and claim that the very concept of "center" has a metaphysical meaning that is imposed on the world. Let's say you are 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" states of a body that does not have a center, this is irrelevant to physics? In other words, 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 towards its center and disappear from the world) then there would be no such bodies. But there is no such law.
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