Math and Reality

[Purple Rabbit]

"Pure mathematics consists entirely of such assertions as that, if such and such a
proposition is true of anything, then such and such another proposition is true of that
thing. It is essential not to discuss whether the first proposition is really true…. Thus
mathematics may be defined as the subject in which we never know what we are talking
about, nor whether what we are saying is true."
Betrand Russell

"How can it be that mathematics, being after all a product of human thought which is
independent of experience, is so admirably appropriate to the objects of reality? Is
human reason, then, without experience, merely by taking thought, able to fathom the
properties of real things?"
Albert Einstein

In the first part of Roger Penrose’s book The Road to Reality Penrose argues that mathematics has an objective reality. He points out that everybody who studies math comes to the same conclusions, and that seriously studying math has the feel of uncovering deeper and deeper truths.

It is undeniable that math is an investigation shared by the human race across time. However, this does not mean that math has an objective reality beyond its existence in humanity’s shared knowledge. Penrose belongs to a school of thought that is quite large in the world of math and that most attribute to Plato: the school that mathematical concepts have their own reality. A reality separate from nature that somehow dictates nature. IMO this borders on supernatural beliefs and as a naturalist I feel I should be able to disentangle the argument Penrose stipulated.

This is a subject on which I am in the phase of forming opinion. I might argue against it and then again I might argue in favor of it. Please join me on this quest for truth.

[Violet]

Interesting idea... I honestly had no idea what you and Adrian were discussing at first.

Why would the reality have to be separate from nature? And is math somehow different from any other thing a girl can think?

If you can make the argument that math is 'supernatural' or 'objective'... then you can make the same argument with any other thought. We know that thoughts are a product of our brains... so they probably aren't supernatural. But are thoughts (concepts rather?) objective? That is a harder question i think.

A man born blind cannot see blue, but that doesn't change the properties of anything. He can't even know what blue means... the idea is beyond his ability to grasp. And so every idea is the product of firstly what we perceive, secondly what we infer from there. A small child for the first time sees a person, then sees another person... it probably just began to establish what 1 and 2 are (among other things) from that data.

My position is that math, logic, and the like are so good at determining reality only because they are created to define and describe things we have perceived (we have probably never perceived an apple becoming 15 oranges of its own accord before our eyes... hence the A=A rule of logic. It may ultimately be false... but for what it describes it appears to be true).

So I think that math is natural and "objective"... just as blue exists wether we think about it or not... so do the patterns exist that we call mathematics. I'm also not conclusive on this, it's just what i think so far.

[Purple Rabbit]

Thanx for your reaction Sae in these slums of the forums avoided by the great mathematical minds of AF.

I was intrigued by your statements in the shoutbox, especially this one :

"The concepts do not require the human mind to exist. Only in identifying is the mind required. They are helpful in describing reality because they are reality (or at least the pattern[s] by which it follows)."

That first sentence strongly suggest that mathematical concepts exist independently of a mind that thinks 'm. But what follows suggests that you think that math is a conceptual implementation of what the human mind sees in reality.

That would necessarily have to mean that math 'follows' reality i.e., the pattern you speak of is first somehow observed in nature and then abstracted into mathematics. But that imo is not the case with many mathematical concepts. Complex numbers come to mind, infinities too, Minkovski space was there before Einstein made use of it, and a whole lot of contemporary math has no known correlate in reality whatsoever. So this poses a real problem for the view that mathematical concepts are derived somehow from reality. In fact it looks just the other way around. Many mathematical concepts probably will never have known correlates in reality.

So this is a strong argument for mathematical idealists, which in fact is a form of dualism.

Furthermore in contemporary physics the situation has even 'deteriorated' much more, in the sense that where physical intuition has lead the way to new breakthroughs it has been replaced by math itself. The Nobel Prize winning physicist and one of the fathers of quantum mechanics Paul Dirac was famous for his insightful advice, “follow the math.” Indeed, following the math and leaving our physical intuitions behind has become a necessary and revealing trend in physics, string theory being a exceptionally strong case for that.

[fr0d0]

Interesting idea... I honestly had no idea what you and Adrian were discussing at first.

That's strange as you can check the shoutbox history just like I can. PR disputed the 0.999... = 1 mathematical fact just like you and I did, for our different, perhaps, reasons. So amusingly, here we are all together in the aftermath of PR's fabulations.

PR also disputes the shorthand method and would insist nothing could be correct but an infinitely notated succession of 9's ...hence Adrian's error... correct me if I'm wrong Rabbit.

That isn't to say that there isn't philosophical mileage in considering mathematical concept. Please carry on.

[Purple Rabbit]

Well let's move beyond boring math tricks and head for the real stuff, what is math really?

fr0d0 can I hear your opinion on that?

[fr0d0]

I wouldn't dismiss math. I find it very interesting and contributory rather than trickery or something to be ignored.

I like the notion that math might describe something other than what is evidenced naturally. Pure logic disconnected from reality should precede reality. If only because the basis of logic is derived from what is observed. And then it is also limited by predefined laws. Science then comes to the rescue and by observing actual reality ever more finely and accurately both intentionally and unintentionally.

I'd love to talk about personally and voice and hear thoughts more clearly than I fear is possible here.

[Violet]

Thanx for your reaction Sae in these slums of the forums avoided by the great mathematical minds of AF.

I was intrigued by your statements in the shoutbox, especially this one :

"The concepts do not require the human mind to exist. Only in identifying is the mind required. They are helpful in describing reality because they are reality (or at least the pattern[s] by which it follows)."

That first sentence strongly suggest that mathematical concepts exist independently of a mind that thinks 'm. But what follows suggests that you think that math is a conceptual implementation of what the human mind sees in reality.

That would necessarily have to mean that math 'follows' reality i.e., the pattern you speak of is first somehow observed in nature and then abstracted into mathematics. But that imo is not the case with many mathematical concepts. Complex numbers come to mind, infinities too, Minkovski space was there before Einstein made use of it, and a whole lot of contemporary math has no known correlate in reality whatsoever. So this poses a real problem for the view that mathematical concepts are derived somehow from reality. In fact it looks just the other way around. Many mathematical concepts probably will never have known correlates in reality.

To use the color of blue as an example... it is just a hue, but upon perceiving it we turned that hue into a concept, and later into a word. Logic (upon which math is based) has been continually observed in nature... that we can use that base to consider things we haven't even seen yet is due to our trust in logic. If logic is how reality "follows" (at least "here"), then what is based off of it is probably true. However, if math is wrong at a point, it would only be because logic does not apply to the scenario. Mathematical concepts are derived from logic, which is an unprovable assumption we've made to reality. It may not be the "true" reality, but it is the reality as we can see it.

So this is a strong argument for mathematical idealists, which in fact is a form of dualism.

Furthermore in contemporary physics the situation has even 'deteriorated' much more, in the sense that where physical intuition has lead the way to new breakthroughs it has been replaced by math itself. The Nobel Prize winning physicist and one of the fathers of quantum mechanics Paul Dirac was famous for his insightful advice, “follow the math.” Indeed, following the math and leaving our physical intuitions behind has become a necessary and revealing trend in physics, string theory being a exceptionally strong case for that.

If math is ultimately a representation of reality, then following the math would be following the reality. For all we know, infinites might be real (see singularities?). Logic itself is an assumption made... but it remains that it could all be one outlandishly ridiculous coincidence. Following the math would be entirely logical given that it hasn't really let us down yet when describing, and we can see no reason for it to do so. But all of math blows up if the illogical starts occurring. Logic is simply an observation we've made of 'reality'... and there is no way we can test it's validity without using itself.

[TruthWorthy]

Math is a concept based on physical reality. It's both a theorhetical construct and logical system of derived rules.

[tackattack]

I'd see the last part of PR's notes in post 3.

I personally don't feel math is based off of reality. It's based off logic and follows reality usually. It's an abstract logical system. Is it relative or proof of reality? No. Is it indicative of percieved reality.. yup. My 2 cents.

[Rhizomorph13]

I think math is just a language to express truths about reality. Sae speaks of blue being a real hue that, when perceived, becomes first a color to our eyes (if we can see) then some precursor impulses in our brains that becomes the word "blue" after passing through our speech centers. I would say that blue is a wavelength of light. The word blue happens to be a great vehicle to pass the knowledge to another person but is no more the color than our perception is the color. It is possible that when I see blue Sae perceives what I would call red even though she would agree that it is blue. This might explain why we all like different colors, and combinations of colors, because they appear differently depending on how our person formed.

So does math = reality? Only as much as any other descriptive language equals what it is describing. We only ever perceive a model of reality, and out of necessity we have to assume that there is an objective reality.

Rhizo