Does Math Exist Independent of Our Minds?

james blunt said:

The complexity of numbers that represents quantifiable values exist in our minds as ingenuity representation of process . The processes exist independent of our minds thus requiring no existence of math to be that process .

Click to expand...

In what form do they exist independently of our minds? They're abstractions, no? And the only place we find abstractions as such is in working brains, no?

blü 2 said:

Hmm. On the one hand, clocks have been modular since the 15th century, and devices for counting fixed repetitive quantities work on a modular basis. On the other hand, it took one of the pre-eminent mathematicians of the ages, Gauss, to turn it into a body of highly abstract maths.

At that level the line between invented maths and discovered maths has more or less vanished, as the better mathematician goes looking for ways to express ideas that come to her when contemplating the maths that already exist. Have you read Simon Singh's highly readable "Fermat's Last Theorem"? It follows Andrew Wiles through the whole creative adventure of marshaling known maths and devising new bridges between various parts to crack the Fermat riddle. Again, invention or discovery, or no clear line between the two?

Click to expand...

Read?
A book?
Nah.

Revoltingest said:

Read?
A book?
Nah.

Click to expand...

Check Amazon and see if there's a bande desinée yet.

Skwim said:

Right! there is nothing to perceive them or label them. However, the quantity, "the amount or number of a material or immaterial thing not usually estimated by spatial measurement." of eight would still exist.

Consider:
Setting aside the questionable status of Pluto, in our solar system there are eight major celestial bodies moving in elliptical orbits around a star. Taking the planets as representing a quantity of eight, this quantity has a mathematical relationship with the quantity of stars in the solar system, which is one. So the relationship is 8:1. And, this relationship holds true whether there's a human mind, or any kind of mind, in the universe or not. So the 8 and1 have values aside from any perception of them.

Click to expand...

The various forms of matter (energy) would exist, the quantity and designation of "eight trees'" would not.

David T said:

state that mathmatical realism is false? Hahahaha.

Click to expand...

So apparently you didn't arrive at your belief of mathematical anti-realism by any logical process. I don't suppose you could tell us how you did arrive it? Was it by looking at cartoon drawings such as you are so talented at posting?

Nous said:

So apparently you didn't arrive at your belief of mathematical anti-realism by any logical process. I don't suppose you could tell us how you did arrive it? Was it by looking at cartoon drawings such as you are so talented at posting?

Click to expand...

Mathematical realism is the view that the truths of mathematics are objective.

Delusional. sorry gollum is not "real." apparently you have zero experience in 3D. Dont confuse the model with reality nature does t work that way. If it did, only mathmaticians would have any real access to reality. Thats a self deluding joke reinforced collectively like a cult.

Sunstone said:

Does math exist independent of our minds? Why or why not?


In the philosophy of mathematics, there are basically two positions on the subject of whether or not mathematics exists independent of our minds.

Naturally, these positions date back over 2,000 years to the ancient Greeks. The first position is called Platonism, after the famous Greek philosopher, Plato. Essentially, Platonism holds that mathematics exists apart from out minds.

On the other hand, the second position -- called Formalism -- holds the opposite. Mathematics is a construct of our minds.

What think you?

Click to expand...

mathematics is a logical framework, of sorts. It is a way for human minds to describe phenomena. The phenomena themselves would exist independent of human minds. The math would not.

PureX said:

The various forms of matter (energy) would exist, the quantity and designation of "eight trees'" would not.

Click to expand...

I take it then that you believe that matter---physical substance---only exists because of our comprehension of it. That when the last sentient mind dies the material universe blinks out of existence. Interesting.


.

exchemist said:

I am not sure these positions are mutually exclusive.

Humanity can create intellectual artifacts that are abstract. For instance, does Bach's Art of Fugue exist when it is not being performed? I would argue that it does - and not merely in the notes on a manuscript used to record it.

So I would say mathematics is a human construct of logic but, having been created, it exists in the abstract, independent of whether any given human mind may be using it.

Click to expand...

Nature has attributes that may described in terms of Math, and humans discovered nad constructed a logical framework for these math attributes and human math was initially functional in describing nature and useful in everyday life and science. Math over time has become more sophisticated with the development of abstract math theory without an intent to be necessarily useful and descriptive of nature and in science, and some abstract math have in some cases been found useful for utilitarian purposes, and descriptive of nature.

shunyadragon said:

Nature has attributes that may described in terms of Math, and humans discovered nad constructed a logical framework for these math attributes and human math was initially functional in describing nature and useful in everyday life and science. Math over time has become more sophisticated with the development of abstract math theory without an intent to be necessarily useful and descriptive of nature and in science, and some abstract math have in some cases been found useful for utilitarian purposes, and descriptive of nature.

Click to expand...

I suppose that may be historically correct, although I would suggest even ancient Greek geometry was already abstract.

exchemist said:

I suppose that may be historically correct, although I would suggest even ancient Greek geometry was already abstract.

Click to expand...

I consider Greek geometry to more on the practical application as well as discovering the geometry in nature for example the golden mean geometry. Also practical application in Greek architecture and design.

shunyadragon said:

I consider Greek geometry to more on the practical application as well as discovering the geometry in nature for example the golden mean geometry. Also practical application in Greek architecture and design.

Click to expand...

That's interesting, because it seems to me that the way they thought about geometric figures was essentially abstract. The very definition of a triangle, for example, is an abstract thing, as are the various properties and relations they established for geometric constructions. Do you think they worked on the the theorems of geometry with applications in mind? I'd have thought not, myself. I think it would have been pure mathematical curiosity.

This article on Euclid's Elements, Euclid's Elements - Wikipedia , for instance, suggests to me that the genius of Euclid was the act of abstraction: defining abstract forms and the rules that apply to them, independent of any application.

Polymath257 said:

Sorry, but Platonism isn't very convincing to me. In fact, from what i can see, Platonism is one of the fundamental mistakes of western philosophy. I prefer Hume.

Click to expand...

Granted, but one cannot deny its appeal even today among something like 80% of mathematicians (a statistic surmised by the anti-Platonist Derek Abbott, Professor of Electrical and Electronics Engineering at The University of Adelaide in Australia, in 2013) and a sizeable number of theoretical physicists (as opposed to physicists working in other fields, who tend towards formalism).

Consider this discussion between mathematician Professor Peter Woit of Columbia University Math Department and a physicist reader of his blog:

1. Pete says:
   January 27, 2014 at 3:13 pm
   Regardless of whether or not the MUH is true or not, I was wondering about your view regarding the idea of platonism/mathematical realism in general Pete. I know that a vast majority of mathematicians and a good deal of physicists would adopt a view that mathematical structures and truths are independent of human beings and that mathematical propositions are objectively true/false.
   
   I must admit I think this view is very plausible, considering the history of mathematics and the sciences (especially fundamental physics) and the influence both disciplines have had on each other. This is not to try and bring in mysticism at all, as I am a naturalist and a physicalist, thought the second part gets increasingly harder to penetrate as you decompose “matter” into a collection of atoms that are 99.999% empty space with particles in a nucleus that decompose into further elemental particles represented as mathematical points or “vibrating strands of energy” in String Theory, whatever the hell that would even mean physically.
   
   I mean, when modern physics points to fact that solid, physical matter is in fact vast amount of empty space linked together by interactions of profoundly small particles than seem to have an ephemeral existence all their own, is Platonism or realism about abstract structures and mathematical relations underlying the physical world really so outlandish? I agree it could probably never be tested, making it more philosophical than empirical, but do you think it a reasonable view?
   
   By the way Frenkel’s book, Love and Math, is brilliant so far and its clear from the reading that he, along with a long list of other mathematicians, wholeheartedly embraces the Platonic view without the slightest bit of crackpot “mysticism.”
   
   
   
2. Peter Woit says:
   January 27, 2014 at 3:36 pm
   Pete,
   I myself favor some form of “Platonism” or realism about mathematics (see a following posting, which has a link to something about the “Putnam-Quine indispensability thesis” and I think Quine had some of the most to the point things to say about how to think about what is “real”).

Woit is a well-known critic of speculative, untestable ideas in science (he is perhaps the most famous critic of the multiverse and string theory), yet even he adheres to a variant of mathematical platonism i.e.


Towards a Grand Unified Theory of Mathematics and Physics by Peter Woit




There is obviously something motivating this phenomenon of mathematical realism that goes beyond a mere philosophical predilection on their part.

I myself was rather surprised to realise just how many researchers in the aforementioned disciplines adhered to a form of Platonism. Its not what I would have expected.

But, Platonism is alive and well.

I wrote: "The various forms of matter (energy) would exist, the quantity and designation of "eight trees'" would not."

Skwim said:

I take it then that you believe that matter---physical substance---only exists because of our comprehension of it. That when the last sentient mind dies the material universe blinks out of existence. Interesting.

Click to expand...

How you "take it" has nothing whatever to do with what I wrote. If you can't read and understand english, we'll have to stop, here.

Vouthon said:

Granted, but one cannot deny its appeal even today among something like 80% of mathematicians (a statistic surmised by the anti-Platonist Derek Abbott, Professor of Electrical and Electronics Engineering at The University of Adelaide in Australia, in 2013) and a sizeable number of theoretical physicists (as opposed to physicists working in other fields, who tend towards formalism).

Consider this discussion between mathematician Professor Peter Woit of Columbia University Math Department and a physicist reader of his blog:

1. Pete says:
   January 27, 2014 at 3:13 pm
   Regardless of whether or not the MUH is true or not, I was wondering about your view regarding the idea of platonism/mathematical realism in general Pete. I know that a vast majority of mathematicians and a good deal of physicists would adopt a view that mathematical structures and truths are independent of human beings and that mathematical propositions are objectively true/false.
   
   I must admit I think this view is very plausible, considering the history of mathematics and the sciences (especially fundamental physics) and the influence both disciplines have had on each other. This is not to try and bring in mysticism at all, as I am a naturalist and a physicalist, thought the second part gets increasingly harder to penetrate as you decompose “matter” into a collection of atoms that are 99.999% empty space with particles in a nucleus that decompose into further elemental particles represented as mathematical points or “vibrating strands of energy” in String Theory, whatever the hell that would even mean physically.
   
   I mean, when modern physics points to fact that solid, physical matter is in fact vast amount of empty space linked together by interactions of profoundly small particles than seem to have an ephemeral existence all their own, is Platonism or realism about abstract structures and mathematical relations underlying the physical world really so outlandish? I agree it could probably never be tested, making it more philosophical than empirical, but do you think it a reasonable view?
   
   By the way Frenkel’s book, Love and Math, is brilliant so far and its clear from the reading that he, along with a long list of other mathematicians, wholeheartedly embraces the Platonic view without the slightest bit of crackpot “mysticism.”
   
   
   
2. Peter Woit says:
   January 27, 2014 at 3:36 pm
   Pete,
   I myself favor some form of “Platonism” or realism about mathematics (see a following posting, which has a link to something about the “Putnam-Quine indispensability thesis” and I think Quine had some of the most to the point things to say about how to think about what is “real”).

Woit is a well-known critic of speculative, untestable ideas in science (he is perhaps the most famous critic of the multiverse and string theory), yet even he adheres to a variant of mathematical platonism i.e.


Towards a Grand Unified Theory of Mathematics and Physics by Peter Woit

Towards a Grand Unified Theory of Mathematics and Physics

Peter Woit

Department of Mathematics, Columbia University [email protected]

Wigner’s “unreasonable effectiveness of mathematics” in physics can be understood as a reflection of a deep and unexpected unity between the fundamental structures of mathematics and of physics. Some of the history of evidence for this is reviewed, emphasizing developments since Wigner’s time and still poorly understood analogies between number theory and quantum field theory.

In this essay I’ll argue that unified theories of fundamental physics are closely linked with some of the great unifying structures that mathematicians have found to underlie much of modern mathematics. This can be taken as evidence of a possible “grand unified theory of physics and mathematics” and motivates the search for a deeper understanding of the known points of contact between the two subjects.

The lesson drawn here from history is that the fundamental laws of physics point not to some randomly chosen mathematical structure, but to an exceptionally special one, requiring a deep understanding of the mathematical world in order to fully appreciate it. New understanding of relations between mathematical ideas and ideas in fundamental physics can lead to progress in either field.

Wigner’s “unreasonable effectiveness” miracle is ultimately a claim that a unity of mathematics and physics exists despite our lack of any good reason to expect it. We may not deserve to be part of this miracle, but we can and should continue to try and understand it.​

There is obviously something motivating this phenomenon of mathematical realism that goes beyond a mere philosophical predilection on their part.

I myself was rather surprised to realise just how many researchers in the aforementioned disciplines adhered to a form of Platonism. Its not what I would have expected.

But, Platonism is alive and well.

Click to expand...

And I certainly understand the impulse towards Platonism in math. As I said before, when doing math, it can *feel* like you are looking for something 'out there'. On the other hand, I get a similar feeling when playing any type of strategy game: that the proper move is 'out there'; I just have to find it. Again, that feeling doesn't negate the fact that the game itself is invented by humans.

Looking over the Woit article, it is impressive to me how *little* of math actually impacts on physics and vice versa. There are parts of differential geometry and the (sub-)theory of Lie groups. That leads into representation theory, which mathematically relates to number theory. While these are all 'big ideas' in math, the relatively limited number of them suggests the interplay is more a limitation on how humans think rather than a deep connection between math and the real world. At least, that's what comes across to me.

But, probing deeper brings up the real issues: we can ask, if we are Platonists, whether the Continuum Hypothesis is 'really true', as opposed to just being true in some models and false in others. This is a statement about the possible sizes of infinite subsets of the real numbers and is a question originally asked by the discoverer of different sizes of infinity: Cantor. It was proved by Godel (a Platonist) that CH is consistent with the other axioms of math: we can assume it and no new contradictions will arise. Later, Cohen showed the same about the negation: we can assume CH is *false* and that assumption also adds no new contradictions to our system

So, the upshot is that we can either assume CH is true or we can assume CH is false. BOTH assumptions are consistent with the other axioms (although not with each other).

Now, I can go further and say that I can see no possible way that either CH or its negation will *ever* impact a prediction in physics. This is simply not a question that has relevance to anything that physics will ever be able to measure (sizes of infinite subsets of the reals when all measurements have error bars).

So in what possible way is CH either true or false? Does it even make sense to say it has a truth value? What would a Platonist say?

And I can go further. Modern set theory has a host of independent statements that can be either assumed or denied. NONE of them have any conceivable impact of physics. So the Continuum Hypothesis is very far from being unique in this way. Each one of these statements could, conceivably, produce a 'branching' of math: one branch leads to certain results, the other to different results. All results follow from their particular system of axioms. All give perfectly valid formal systems.

This is why Platonism loses in my mind: it simply doesn't have a good way to deal with the known fact that any formal system has such independent statements.

PureX said:

I wrote: "The various forms of matter (energy) would exist, the quantity and designation of "eight trees'" would not."

How you "take it" has nothing whatever to do with what I wrote. If you can't read and understand english, we'll have to stop, here.

Click to expand...

Got you cornered, did I. I figured you'd eventually come to realize your untenable position and have to bail. But that's okay. I understand, and I'll stop.


Have a good day.


.