Philosophy of Mathematics


#41

Agreed. Complex numbers and their placement on the s plane not only determine the stability of the system, but are sometimes detrimental to the stability of the engineer as well.


#42

I can’t place the citation, but it’s been a common saying that mathematics is the language of the universe. And if that is so, I can’t see how we can reason it is not of God.


#43

Ooooh, love Numberphile. Thanks for sharing!


#44

He said it on Hello Internet


#45

No.

Abstract objects can only be abstracted in the mind.

Human discover math.

No, God didn’t create math.

Math is about how the structure of reality can be. 1+1=2 which means in reality one apple plus one apple is two apples.


#46

Would you or someone else like to explain to me why saying “God didn’t create everything because God didn’t create himself” would provoke this response?

Did I offend or am I grossly heterodox for some reason I don’t see or…???


#47

But can’t we say math exists outside of and independent from it’s utility? If you accept that the complicated math that describes the work you do exists, then doesn’t whatever we derive from that truth also exist, however esoteric?

For example, if taking square roots exist and negative numbers exist, I’d argue that then complex numbers HAVE to exist.


#48

I would invite you to engage in discussion further. Thanks for expressing your opinion to get things started.

How can you say humans discovered math if it doesn’t exist outside the mind? You can only discover something that is already there. Unless you’re prepared to claim that math is in the deep recesses of our mind from whence we discover it.

Of course this fails because there are some questions which we are literally incapable of solving despite them having a solution, such as continuum hypothesis. But I don’t really think that’s what you’d say anyway


#49

While I’d say logic is logic in it’s purest form (it’s QUITE a large field) in it’s purest form, I would tend to agree that math is logic. But how does that not come directly to the conclusion I gave in my OP? If truth exists, then it seems the truths about truth exist. Such as: if A then B. A. Therefore B. This HAS to exist because it’s true.

Well it seems to be that if logic implies math (Bertrand Russell did some very convincing work in this area). Then this carries all the way up. Since I’m talking about something confusing let me state it as hyper-formally as possible.

  1. Assume truth exists. (I’m not really prepared to argue about this one for the sake of staying on track.)

  2. Let T = {A,B,C,…} be a set of all true statements.

  3. If we can observe that the elements of T follow some basic laws, then the laws are also true.

  4. Since true, these laws are in T.

  5. The laws exist because they are true.


#50

It might exist before I personally number it. But it seems it is necessarily numbered even if I don’t number it.

How then does the number used to describe it not exist? It seems to exist outside of me and my mind.


#51

Math is about the relation between things in reality. We just realize this “2 apples is more than 1 apple” and then abstract it “2>1”.


#52

I would argue mainly two ways:

  1. How can you recognize that there are two apples if there’s no such thing as “2”. Further, how can you recognize the relationship between 2 apples and 2 oranges? The commonality you distill from the two situations exists outside the mind. 2 apples is the same number of things as 2 oranges whether or not someone takes note. It seems to be the two-ness of the apples is a particular instance of two, which is a general attribute of nature, in this case quantity.

  2. Your analogy of direct relationship falls appart quickly when it comes to more esoteric mathematics. What possible physical meaning can John Conway’s surreal number system have? It might be applicable in some way I admit, but not in a meaningfully direct way. If we can take what we abstract from reality and it leads logically and necissarily to something not reflected in reality then we’ve discovered a truth beyond physical reality. But it seems just as real, derived from the truths reality gives us.


#53

Of how math can be used to describe the world: When all you have is a hammer, everything looks like a nail.


#54

For anyone who thinks that all math describes something about reality, I invite them to pursue a list of famous unsolved math problems, for example, these.

Granted a few of them might have some relationship to reality, but the majority of them do not. For example, the simple but maddeningly baffling Collatz conjecture:

Start with any positive whole number. At each step, if the number is even, divide it by two. If it is odd, multiply it by three and add one. Repeat this process indefinitely. It will always eventually lead to the number 1.

This conjecture has remained unsolved since 1937 when it was first proposed. It is hard to see this conjecture as describing anything about reality. If anyone needs more convincing, just pick a few other problems from the list above and have a go at them.


#55

Oh it’s worse than that. There’s math problems we CANT solve. Look up Gödel’s Incompleteness Theorems. It’s certainly not my claim that all math must describe a physical reality.

My claim is that math exists independently from the physical world.


#56

Negative numbers and square roots are necessary but not sufficient conditions for positing complex (more specifically “imaginary” ) numbers. There is an implied rotation to orthogonality. I have seen some scientists exploit this property in the Quantum Mechanics field.

Does math exist outside of its utility? Maybe. But that speaks, maybe, more towards why it exists, rather than that it exists.


#57

I’m sorry. I thought I was replying to STT. And yes, I know about Gödel. I even had the proof presented to me at one time (on how to construct an undecidable number theory statement) but I can’t recall how it was done now.


#58

What it would mean for math to exist only in it’s utility? This seems to imply “2” doesn’t exist unless two things exist.

And hm. You do make an interesting point. But I think the condition is also sufficient. Think about it in reverse. Imagine that complex numbers do not exist. Well it’s hard then to say that i^2 = -1. How can we say there’s any relationship between something which exists and something which doesn’t let alone a truth claim. It seems to me that if truths exist and truths we accept imply other truths, we must accept those too.


#59

Actually, e^i*pi = -1. Euler’s equation. That gets you going down an alternate path to arrive at the equivalent answer. It also pulls in alternate coordinate systems and trig into the mix.

Which makes one think… why should simple operations, performed on two transcendental numbers, and an imaginary identifier produce negative unity? Interesting, no?


#60

Indeed :slight_smile: As a math major (almost done woo). This fact is well known to me. I don’t see exactly how it relates to this conversation, but I’ll nerd out about Euler’s identy if you wish lol.

My FAVORITE part about it is that it allows us to analytically continue that natural logarithm. As then ln(-1) = pi*i (on the principal branch).

An even more amazing fact perhaps, is that i^i is a purely real number.


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