I gave a concrete example from the realm of negatives.

As stated, what interests me is the question of what the rules are for translating mathematical language into factual claims about the world. My basic intuition says that in order to assume that a certain mathematical model has a counterpart in the real world, we need to show how it can be converted into a verbal statement that has meaning.
The mathematical expression:

2 + 3 = 5

can be translated into the following verbal statement:

"Two oranges plus three oranges are five oranges"

Clearly this is a statement that has meaning, and beyond that we feel that it also “works” in the real world.

By contrast, we already agreed that the mathematical expression:

9 – 7 = -2

if translated into a verbal statement, is meaningless, and so we would apparently conclude that it has no counterpart in the world.

So far I’ve brought simple mathematical examples for which it is relatively easy to decide whether they apply to the real world or not.

But there are much more complicated questions. For example, what is the status of the mathematical model that describes a world of 10 dimensions (string theory)? Even if we assume the mathematics there is completely “kosher,” it is still not clear whether it has a counterpart in the real world.

Another example that popped into my head is the “ontological” status of the square-root function.

I’m not asking you to address all the examples, of course.

What about non-Euclidean geometries?

To conclude, I propose the following thought experiment that might help “approach” my question:

Let’s imagine an adult who has mastery of mathematics. By some miracle, language as a whole has been completely erased from his consciousness (and even his thinking has ceased to be verbal).
Would such a person be able to handle mathematical problems exactly as before?

In addition I’ll bring more examples.

You’re not talking about translation/conversion but about application (in mathematicians’ language: the relation between a theory and a model). The claim -2 = 7 – 9 has an excellent application, and I already gave it: you remain in debt by two oranges.
And of course this has nothing whatsoever to do with our question of language. Our language describes mathematical entities just as it describes entities in the world, so the question of the relation between mathematics and the world has nothing at all to do with language.
In short, again I see here a meaningless question.

You cheated the moment you said, “A person has seven oranges.”
That’s not true at all.
A person has oranges. And he counts seven.
Counting is a mental operation. It’s not something he has.

I’ll bring up a concept from mathematics that doesn’t exist in reality: complex numbers.

The number 3 doesn’t exist in reality either. In reality there are sets of three objects, and the number 3 describes them. There are also things that are described by complex numbers.

Decisor,

Clearly the act of counting is a mental operation.

My question was about the relation of that operation to extra-mathematical reality. If we go with your approach, according to which a person doesn’t really have (outside the mathematical expression) a given number of oranges, then we have thereby turned mathematics into a completely arbitrary domain. We could arrive at any result we want, for example that there are 300 oranges there. Everything is mental and “all in the head.”

Michi,
I’m unable to understand your claim.

My assumption is that both mathematics and language carry some information about the same object of reference (in my example the first object is the negative-number exercise and the second object is the simpler exercise with ordinary positive numbers).

As long as we haven’t gone beyond the bounds of the mathematical expression, there’s no problem at all: both the negative-number exercise and the positive one are “kosher.” But the moment we transfer (apply) that same exercise to the linguistic-verbal medium, the fates of the two expressions diverge: one still carries meaning and is “valid,” and the other does not.

From this problematic I identify at the very least an open question (understandable and clear): what makes one expression applicable in language and prevents the other from having that same possibility?

Michi,

Now I see that you also claimed that the negative-number exercise has an application: the concept of “debt.”

I’m processing your answer and I’ll check whether it works for me.

The relation is between the mental operation of counting and the mental operation of perceiving the orange.
We are always in the domain of perception.
And there is no problem saying that we perceive the world in such a way that the mental mathematical operations fit it.
In other words, there is a mental connection between the way the world is perceived and mathematics. Then it isn’t arbitrary.
But one should also remember that that same act of counting, which is the basis of mathematics, is only a small part of world-perception.
For example, the perception of the orange’s color cannot be described mathematically.

If something works (has an application), then that means it is true. And among other things that means that the entities on which it is based are real and alive and existent. Yes, including complex numbers. It really is a wonderful phenomenon in science how ad hoc gimmicks and mathematical aids over the years become something like a natural phenomenon for physicists (one of the famous examples is the electrostatic field, which turned from an auxiliary tool for calculating force—and maybe the magnetic field too—into a physical entity of equal status to material particles: the electromagnetic field). Don’t forget that for similar reasons, zero too was once not considered a number. One could just as well say that the natural numbers don’t exist either.

That is, the separation we make between the physical world and the world of abstract entities is not sharp. There is a continuous transition between them. In that sense, a monetary debt (which is represented by a negative number) is an existent entity no less than the chair I’m sitting on.

It seems to me the time has come to wrap this up. 🙂

Small correction: I meant that a monetary debt (which is represented by a negative number) is an entity no less existent (and not tangible, as I wrote) than the chair I’m sitting on.

Decisor,
I’m not sure you solved the problem this way, but only pushed it back. How will you now know that the perception is of three oranges and not 300?

Michi,
The move from the expression:

-2

to the expression:

a debt of two

certainly indicates some correspondence between them, but fundamentally it creates a completely new meaning. That meaning cannot reside in the original mathematical expression.

Therefore my intuition from the outset was that the difference involved in moving from one medium to another could serve as a touchstone for the question of mathematics’ applicability to the world.

Immanuel,
If you argue that the expression:

"debt"

reflects an existent entity (no less than a chair), how do you want to defend your pragmatist position?

Note that I am not disputing at all that this expression in its verbal form “works.”

Your claim in effect gives priority to a theoretical description of abstract “entities” (of the “debt” type), even if they don’t “work.”

Since Rabbi Michi said that the time has come to end it, then unless I receive his permission I cannot answer (even though I do have what to answer). But the question is whether by this very response I’m already violating his words…. 🙂

Answer us, Immanuel, answer us.
Rabbi Michi meant to end his discussion with Doron (because they had reached a temporary agreement and it’s better to quit while you’re ahead). Not that this thread should stop in general. Because what is the difference between this thread and other threads where other people are discussing and no one told them to stop.

Indeed. On this site everyone is given freedom of speech. I said that I was done.

Of course. I meant that I was done. Everyone can express themselves here on the site as they wish on any subject.

All right,

once permission has been given for the chatterbox to speak, he doesn’t distinguish between righteous and wicked.

So I am not a pragmatist, and in general I have no need to defend positions. Pragmatism is a method in which morality is defined by the benefit in it. That is, good is what benefits society as a whole, and evil the opposite. I believe in good and evil in themselves, only I believe that in the long run the good is also supposed to bring benefit and evil brings harm. That is, I believe in reward and punishment. And if in the long run no benefit comes from a good act, that is a sign (not the essence) that my act was not good in the first place, and it should be reexamined.

As for our matter, the discussion here is not about good and evil but about truth or falsehood and existence or non-existence. So if a concept is fruitful (works), that is a sign (not the essence!) that it is true and exists. And once that is so, since it is more spiritual (more abstract), it also exists more. Just as once it becomes clear to me that an act is truly good, doing it for the sake of the good in it is loftier than doing it for the sake of the benefit that comes from it (even though there is a connection between them, like soul and body).

Correction: “Just as once it becomes clear to me that a certain act really is a good act, then from my point of view doing it for the sake of the good in it is greater than doing it for the sake of the benefit that comes from it (even though there is a connection between them, like soul and body).”

To the question, what is the criterion that distinguishes between mathematical expressions that apply to reality and those that do not, the linguistic criterion was proposed.
As a concrete example, the concept of the negative was proposed, and it was argued that since its translation (application) to the verbal medium is problematic, perhaps one can learn from this that this operation is only a convenient and efficient convention, but does not necessarily reflect real reality.
As a counterargument, Michi proposed “translating” the concept of the negative into the concept of “debt.”
I argued that this answer is unsatisfactory because the concept of debt loads onto its mathematical counterpart (the negative) meanings that simply are not there to begin with.
At this point that specific discussion was cut off.

At the margins of the discussion, pragmatism was discussed, and the mistaken claim was made that it deals only with morality.
For the benefit of anyone interested in a somewhat more faithful-to-reality description of this philosophical school, I’m attaching its entry on the Stanford site, along with a short excerpt from the introduction.
That said, one should not expect the consistent pragmatist to be overly impressed by what is written there if he cannot derive “benefit” from it.

Pragmatism is a philosophical tradition that – very broadly – understands knowing the world as inseparable from agency within it. This general idea has attracted a remarkably rich and at times contrary range of interpretations, including: that all philosophical concepts should be tested via scientific experimentation, that a claim is true if and only if it is useful (relatedly: if a philosophical theory does not contribute directly to social progress then it is not worth much), that experience consists in transacting with rather than representing nature, that articulate language rests on a deep bed of shared human practices that can never be fully ‘made explicit’

https://plato.stanford.edu/entries/pragmatism/

We’re going in circles here. In this context I don’t care about the history of philosophy and who said exactly what and what exactly he meant. What matters is that I understood you and that you understand me, and that’s it. I was talking about the fact that a necessary indication (but not a definition!) of truth is fruitfulness (not because of the benefit I get from it, but by virtue of the fruitfulness itself). That is my observation. And it seems to me that of many others as well. Barrenness is a sign of falsehood. If you don’t see that, then I have nothing to say to you except that you should try to understand why what I’m saying might be correct. Don’t think otherwise because of some desire not to belong to some club.