Is math truly objective?

Great Debates
41

I looked through the sucky search engine, and did not find it. (There are articles I had in print that did not appear there before) I have it in an old issue from the last calendar year. I will look it up. It is not a major feature article, may have been a book review. (no one here read it? hmmm.)

In short the piece talked about new ideas challenging the concept of an infinite number of anything.

The infinite space article raises a good tangent. If math is supposed to be concrete and objective, as a language it must represent a real thing. If the universe is not infinite (an idea not limited to this one article - it is the current big debate in physics), then infinite numbers do not represent anything real. Also, if the spatial limits of the universe are finite, then quantity of matter in it, while large, is not infinite.

42

To see how truly sucky the “search engine” is, try single words like - “number” or “theory”. I was getting zilch. Apparently it is only by tagged files, not exhaustive text, and if the site hasn’t been spidered recently, a domain search would be useless (domain:sciam.com AND “number theory” for example).

My citation was not meant to prove the point, but raise the issue, hopefully someone has read it, or I will continue to dig.

43

I suppose it partially depends on your definition of a real thing. To me, a concept or idea is a “real thing” in and of itself. Even if there aren’t an infinite number of any “things” in the universe, the concept/idea of infinity is still a valid one, and one that can be studied mathematically.

Suppose there are a maximum of n “things” in the universe (n being finite). Does that mean that n+1, as a number, is meaningless, or doesn’t even exist? Even as a concept?

44

Depending on which definition you use Math can be subjective or objective. It is subjective in the sense of existing only in the mind, it is objective in that it is uninfluenced by emotion, surmise, or personal opinion.

The most salient feature of Math is abstraction, making general statements that don’t apply to any external objects in particular. Whether the universe in finite or not has no bearing on number theory. Cantor and Godel weren’t concerned about whether physical objects could be found to support their concepts of infinity, it’s an abstraction.

There is a branch of math called constructivism that insists that only concepts that can be constructed can be said to exist. The result is a different set of rules and theorems, but it is still math. There is room in the world of math for many different camps, that’s one of the reasons it’s so interesting.

This site talks about constructivism and also addresses the rejection of infinite sets:
http://www.britannica.com/bcom/eb/article/2/0,5716,119912+9,00.html

Enjoy!

45

Ideas and constructs are real things. But perhaps not in the same way that physical objects or processes are. Many people believe that God is real in the same way that some believe “inifity” is real. Obviously as this board attests, there is some contention on this issue.

To assert that mathematics is “objective” is a different goal than to assert that the ideas expressed by mathematics are “real”. The objective part would require that any human sufficiently trainined in a type of math would draw the same conclusions, probably to a greater degree than natural languages seems to (English, Chinese, etc.), and to a much greater deal than beliefs, unfounded opinions, impressions or the much maligned intuition does.

To be considered objective, the test is more rigorous than these other schenarios. We might require that mathematical statements depict real phenomena, in order to be able to test whether they are objective.

In the above example of the n things in the universe, adding one more (this lat thing being non-existant) to arrive at n+1 begs the question: does this number depict anything “real” (existant). The answer is no, in terms of the number having any relevance. It does seem to exist as a concept, thus we have a dividing line between where objective math leaves off and speculative math begins. There are many such boundaries, perhaps they are infinite, although for concepts to be generated infinitely, we would need an infinite number of years, civilizations and brains (even non-human will do).

I’ll take the radical unfounded (expect by the piece mentioned above which I cannot track down) that infinity is a concept only, created by human math. Like the earth which approaches flatness at a sufficiently small scale, the universe seems infifnity when viewed on our scale. Not just an empty metaphor, a curved universe is a model that makes a finite universe possible AND allows for straight vectors to retrace themselves once they have gone far enough.

We humans are having a problem here modeling the universe on a grand scale. We need to imagine space like a large ball so that a finite universe still seems unbounded (a circle is unbounded but finite), or that space is linear in some way and has the property of infinity so that the damned end point with nothing beyond won’t be a problem.

If the sub-atomic is as conter-intuitive as we’ve found, no doubt the universe is too. The need for inifinty AND finiteness as concepts are grounded in human experience, and our own scale in the universe.

46

If math is really objective, how come there’s so much of it I don’t understand?

That sounds funny, but it goes to the heart of an intuitive understanding of what it means for something (call it O) to be objective: that is, rational person A can convince rational person B of the truth value of O in such a way that B has sufficient reasons for believing that A is correct. To put it another way: A can convince B that O is true (or false) through rational arguments. That’s why the law of gravity is objective and art isn’t, right?

Now, there are many people who are not good at math and who are incapable, even in principle, of understanding many mathematical concepts. They are not irrational, just not very good at math. Given that they cannot be convinced of the truth of certain mathematical statements, in what sense are those statements objectively true?

Suppose one took the above argument and substituted art for math, beauty for truth, and talked about the objective beauty of art? What does it mean to say something is objective and how would we know if it were?

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They’re objectively true because they’re not subject to that person’s feeling about them - they are provable (given certain axioms). That some particular person may not understand the proof does not change its validity (or lack thereof).

–
peas on earth

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Are you saying that they understand the law of gravity? Einstein’s law of gravity? But not math??
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RM Mentock said:

No, just that they understood the idea in a general way. You know, if you let go of something heavy it will fall, that sort of thing.

bantmof said:

So the situation is something like “95% of the human population could understand X, and the rest should accept is as objectively true, even if they can’t understand why”. What if only 5 people in the world understood the proof? Would it still be objectively true? How would we know they weren’t kidding us? I guess my real problem is epistemological: How could we know something is objectively real? It seems odd that something could be objectively real, yet you couldn’t convince somebody else of that.

50

Yahbut, math is that way too. Most people understand math in a general way. The analogy seems to be apt. I’d think you’d either have to start asking whether gravity is truly objective, or concede that most of math is.
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51

This may be going a little far afield, but what about “renormalization.”

This procedure is used in quantum physics to get rid of infinities that pop up when dealing with the mass and charge of a particle. To get rid of the infinities, physicists put infinities in by hand so that they cancel out.

I’m no mathematician, but this looks very much like sleight of hand to me.

In what way can this particular application of mathematics be said to be objective if values are inserted arbitrarily for no other purpose than to get the answers to come out right?

Is my understanding of this situation incomplete, or is there something weird going on here?

52

I think it would still be objectively true. I don’t know how the rest of us could establish that those 5 weren’t yanking our chain, but the situation seems unlikely to come up. The tricky part is usually coming up with the proof in the first place; once that happens, the body of people who can understand the result is usually far larger.

There are some proofs that are relatively inaccessable just due to their sheer size, but that really isn’t a problem of only a few people being able to understand them - it’s a problem of only a few people actually bothering to do so.

In fact I’ll go one step further; let’s say a valid proof was handed to us by space aliens, and nobody on earth understood it. I believe it would still be objectively true.

–
peas on earth

53

Mathematics is perfectly objective because it is perfectly abstract. It begins to become subjective when it is applied to the real world.

Simple example: What’s 2 + 2? Ok, now what’s 2 cups of rice + 2 cups of water?

Noam Chomsky has a great deal to say about the inapplicability of any pretention to scientific objectivity to social and political fields.

54

If you and I were standing together and I noticed an anvil falling from above towards your head, and I could in no way convince you that it was true, does the anvil not fall on your head?

I think many misunderstood my question, save RobRoy and a few others. BTW, brilliant posts, RobRoy. Perhaps I should reprase the question, “Is mathematics universal?”

I mean, if we were ever to develop the technology to visit other civilizations in other worlds, would they adhere to the same logic as us?

I’m still only a student of mathematics, although I can sort of grasp a “feel” for the whole thing. The number “1” exists as an idea. The fact that “2” is greater than “1” is another idea. From there, with a few axioms concerning operations, we can by induction create the set of natural numbers. And from there the integers, rationals, reals, and complex. Does the number “1” exist independently of human thought? Howabout the axioms we take for granted?

Sorry, I know these are futile questions, but they are very interesting in my drunken moments.

There’s always another beer.

55

Here’s an interesting proof that I stumbled upon:

There exists an irrational number r such that r^(sqr(2)) (r raised to the power of the square root of 2) is rational.

Proof.

If sqr(3)^sqr(2) is rational, then r = sqr(3) is the desired example. If sqr(3)^sqr(2) is irrational, then [sqr(3)^sqr(2)]^sqr(2) = [sqr(3)]^2 = 3 is rational.

So an irrational number raised to the power of an irrational number turns out rational, very interesting.

There’s always another beer.

56

Beeruser

I’m not going to supply a proof, as you did, but here’s a related factoid, concerning a transcendental number raised to an imaginary transcendental power that turns out to be an integer.

e^(i pi) = -1
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*RM Mentock: I’m not going to supply a proof, as you did, but here’s a related factoid, concerning a transcendental number raised to an imaginary transcendental power that turns out to be an integer.

e^(i pi) = -1*

Or, rewriting it:

e[sup]ipi*[/sup] + 1 = 0

This equation lists many mathematical bases and identities:
0 - additive identity
1 - multiplicitive identity
e - natural logarythmic base
i - imaginary number base
pi - trigonometric base

    • addition
    • multiplication
      Exponentation
      = - Equality

What would Brian Boitano do / If he was here right now /
He’d make a plan and he’d follow through / That’s what Brian Boitano would do.

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That’s Euler’s. And it’s the most beautiful thing ever conceived.