Philosophy of Mathematics


#101

Oh boy, we do no such thing. You just haven’t been taught it. Division by zero is defined in a set of mathematics called Wheel Theory. Look it up, you sound interested.

The problem is that this generalization causes a loss of generalization for other properties. Such as the distributive law, the existence of unique multiplicative inverses etc. While it is useful to look at algebraic wheels and see what is true about them, we can prove a lot more if we stick to fields. That’s why people prefer to use them.

It’s wrong, flat out wrong to say “you can’t divide by zero” but mathematicians say it all the time. Why? Because the algebraic system you develop is much more useful. Dividing by zero is of much smaller utility than the distributive law, for example. But it would be more correct to say “you can’t divide by zero, if you want the distributive law to hold”

Edited to add that 1 =/= 2 but only because 0=/= 1 by definition. This is again due to a frustrating loss of generality would result. There are however number systems where 0 = 2 (modulo arithmatic). Or even crazier things like apperently infinite numbers being equal to negative numbers (p-adic numbers).


#102

I do not understand your notation. But I will decline to argue with Aquinas in this case.


#103

Those fields in mathematics that model things in the real world are not arbitrary, but then again, some fields in mathematics are quite arbitrary. For example, in combinatorics, and the search for complete sets of orthogonal latin squares. The definitions of a latin square is arbitrary, and the problem of classifying all of them does not come from the real world. Or take the Collatz Conjecture (google it). It has consumed much effort by mathematicians, but the rules of that problem are clearly arbitrary.


#104

Thank you for the correction. I did not know. I’ll look it up.


#105

There is no such thing as the set of all true statements.


#106

I would reformulate the question: What is the ontological status of mathematics? Ontology is the science of being. Things might have different ontological status according to their form, location, nature, etc. of existence. If something is in your mind only, it still could exist. Your mind is mapped to your brain with nerve cells. Your thinking and ideas are coded into the configuration of your nerve cells. They are represented and stored by little particles of biological material coded/ decoded by your brain at a tiny fragment of a second. This is a physical/ material mapping of everything you have ever thought into the tangible and measurable. Doesn’t this mean some kind of existence? Absolutely does! Mental constructs may acquire high ontological status.

This is the case with both math and God. We are only able to grasp the concepts of math and God through abstraction. Abstraction is a mental construct to describe complex phenomena via sophisticated brain activity. This ability is a uniquely human feature. So as human beings we have high capacities to create things with ontological status. The same time, it is a limitation of assurance of the existence of things. We cannot assure existence beyond our minds. Even the evidence for the existence of God is understood by human brain activity.

This might not be correct. Giving ontological status to mental constructs seems to be a transcendental activity. God is transcendent, so His ontological status is infinite. Mathematics is not transcendent in the same way. First of all mathematics, unlike God, is unable to create matter out of nothing. So the ontological status of abstract mathematics is finite, even though limitless.

By the way, how would you attach existence to a nice concept from mathematical analysis like the uniform convergence of continuous functions of a closed interval?


#107

There are ground rules of a specific problem and there are rules governing mathematics.
The ground rules of the Collatz problem are not the rules governing mathematics.


#108

In a way, the only ground rules governing the study of mathematics is logic, which one would have to agree is not arbitrary. But the things that mathematics studies are arbitrary objects. The Collatz Problem is one such thing that is studied. It is not unlike people who study the game of chess. It is true that they are bound by logic. But the rules of chess seem quite arbitrary.

The discussion earlier was whether mathematics had any “existence” outside of man’s invention or discovery of it. One can similarly ask if chess would have any existence in a world where no one happened to codify the rules of chess.

So while the ground rules for the study of mathematics are the rules of logic, some of the things that mathematicians study have no existence prior to being made up by someone.


#109

I think this is wrong, I believe in any mathematical system such as number theory or set theory, the set of all statements is countably infinite, so there is certainly a set of all true statements. Now, we could not define whether any arbitration statement is a member, since there are true statements that cannot be proven, but the set exists.


#110

Have spent my whole life doing applied math.


#111

That is a good way to phrase the question. For something to exist, except for God, it had to be created. God’s creation consists of the physical universe, which math is not part if, and the angels in the spiritual realm, which again math is not apart of. Is not math just a form of language, ie a means of description nd communication?


#112

Right, so could it be that math is of God? I admit it sounds a little strange. But it makes sense to me that God is logical. If God is truth and has perfect self knowledge, to me that heavily implies logic. Lots of mathematicians believe math is reduceable to logic. So is God math?


#113

I see what you’re saying. I agree in some sense but not ultimately. By exist I guess I’m saying ‘exists outside the mind’. If something exists only inside the mind then it’s subjective. Well if something only exists subjectively I feel inclined to say “I don’t care that it exists. It may as well not”. It will only exist in the exact state it has in someone’s mind which is unknowable to anyone else.

Such existence is near meaningless to me and by my personal choice of what existence is I’d rather consider that not really existence at all. You can make a different choice but then we aren’t talking about the same thing.

I am not a radical physicalist. I think brain chemistry and neuroscience cannot sufficiently explain why human brains alone abstract. Thus the evidence of God is not ‘in the brain’. And neither is mathematics.

I don’t quite get what you’re asking. Uniform continuity is an attribute of a sequence of functions. If the squence of functions exist it makes sense to talk about its attributes as if they exist.


#114

I have already stated my objection to the phrase “a set of all true statements,” so let me use “the set of all possible statements that are true,” since that is what I think you mean. It expresses more clearly the openness of the set, though if you already have the set of all statements, that is not a problem. Gödel or Tarski fits in here somehow, incomplete or undefinable, but I don’t recall which.


#115

No, I certainly mean “the set of all true statements”, not “the set of all possible statements that are true”. I actually am not for sure about the latter. The former exists, at least I cannot think of a reason why it would not. For any given mathematical theory, there are a countable set of statements. Each statement is either true or false. So there are certainly two subsets, the set of true and the set of false statements. Godel just tells us that there are statements which we cannot determine which subset they belong (using only that theory).

I have reread your posts and I don’t see your objection, could you please clarify?


#116

Is the set containing one or more statements, itself a statement?
Also, in your definition of statement, do you allow for statements about sets of statements?


#117

First answer, in set theory is no, or at least it seems to be an incomplete statement. Its kind of like an in English saying “the boys”, you did not make a statement.
There are certainly statements about sets of statements. This is effectively what Godel formalized when he defined the Godel number, a means of identifying each and every possible statement with a number, and then you can define sets formally that are sets of statements and make further statements about them. It was one of the steps in saying “there are statements which are true, but cannot be proven” and expressing that concept in an actual number theory statement.

note: hope I didn’t get those details wrong, I am working off long, long term memory.


#118

Well then the number of statements is not countable since statements about statements would involve a power set and Cantor’s theorem shows that the power set of a countably infinite set of statements is uncountably infinite.


#119

Who cares if it’s an uncountable set? It probably is.


#120

I envy your optimism. But a set of all sets does not exist either.


DISCLAIMER: The views and opinions expressed in these forums do not necessarily reflect those of Catholic Answers. For official apologetics resources please visit www.catholic.com.