What's Your Philosophy of Mathematics?

[Evo]

I am re-opening this thread under the condition that the thread return to specific discussion of the topic.

 

[chiro]

One thing about mathematics is that, in my opinion, we are simply re-discovering what already existed and what was always possible from the very beginning.

Some people might interpret this to mean that since humans don't actually 'invent' anything but rather just 're-discover' it, that we are not really that special in the fact that we didn't 'invent' something per se. In the above case, if people are thinking that way then are placing way too much importance on the importance of self and are missing the point completely.

In this sense, the above might correspond to a form of Platonism with respect to the above rant.

 

[Evo]

Please stop the nonsense posts.

 

[dimasalang]

i like the way we used the symbols of math by using logic behind it.



nature don't need an intuitive conscious mind, like us to understand itself., were just merely a helpless creation of infinitely small particles. govern by incomprehensibly and mind boggling laws of nature.

 

[Ken G]

One thing about mathematics is that, in my opinion, we are simply re-discovering what already existed and what was always possible from the very beginning.

Would you say that the painting "Mona Lisa" was always possible from the beginning, and does that imply that Da Vinci "rediscovered" it when he painted it? Or is mathematics demonstrably different from art in terms of which comes from us and which is built into nature? Can you trace the details of how each is done to reach this conclusion using pure logic, or do you just have to assume it as a postulate?

I would argue that any attempt to make the above claim true is fundamentally circular-- it can only be true if it is assumed to be true from the outset. So the opinion being expressed is that we should just assume the above. I'm not sure that simply assuming away what we don't actually know is the best way to make progress in philosophical inquiry, isn't a better question, what do we get if we assume that, and what do we get if we don't? Isn't it true that both approaches give us a different perspective on what mathematics is, and don't we want to see mathematics from all valid perspectives? This is the "nonsense" I've been arguing: we are incorrect to imagine that mathematics is only one of the things on that list, for all that amounts to is putting on blinders about what mathematics quite demonstrably is and can continue to be, if we just don't don those blinders.

 

[bohm2]

Or is mathematics demonstrably different from art in terms of which comes from us and which is built into nature?

I tend to think that both comes from us but it seems mathematics is much more useful as a scaffolding to attach our claims about physical systems. It seems that there is something more to physical reality (or even our models of physical realty) over and above the mathematics. It seems that the mathematical theories/objects are not the same type of entities that appear to exist in the physical world. We can't get to the physical world without using mathematics because non-mathematical versions of scientific theories just seem to be practically very difficult to do. But, even though the mathematics may be indespinsible and the mathematical equations we use ultimately decide what we believe about the physical world there still seems to be this difference between the two and this just adds fuel to many of the interpretative debates in science, I think.

 

[micromass]

Please stop the nonsense posts. This thread is about mathematics and its philosophy, it's not about the Mona Lisa nor about determinism nor about the existence of reality, etc.

There are a few interesting points which are touched on, but there is also quite a lot which is not relevant to this thread what-so-ever.

Sure, short analogies are allowed. But as soon as you're saying more about about the analogies than math, then it becomes nonsense. For example, typing 50 sentences about the mona lisa and then saying that mathematics is the same thing, is not allowed.

Please make sure your posts are actually about mathematics, and not about something else.

You've been warned three times now. If it doesn't stop, then more serious actions may be taken.

 

[Ken G]

Please stop the nonsense posts. This thread is about mathematics and its philosophy, it's not about the Mona Lisa nor about determinism nor about the existence of reality, etc.

Art is just a useful comparison, and ontological issues are relevent, for they are referred to in the OP itself (just look at choices #4 and #5, which directly connect mathematics to things that actually exist, i.e., claim that mathematics is explicitly ontological). I understand that you must decide what elaboration is off topic, and I might not agree with you, but it's your place to do that not mine. All I'm saying is that this whole thread is about whether mathematics is something inherently semantic and ontological (connecting with meaning in the real world, choices #4 and 5), or something inherently syntactic and epistemological (a kind of procedure for knowing that follows a fixed set of rules and has nothing to do with anything outside those rules, expressed in all the other choices). Since I am not allowed to actually support my stance, I can only state it: neither of those answers could possibly do justice to what mathematics actually is, because what mathematics actually is is a juxtaposition of those two possibilities. Mathematics takes on its meaning in the place where those things come into contact, because neither of them have any value on their own, they are both vacant notions until they are juxtaposed. To actually describe my reasoning might seem verbose, or might need to bring in how art and language also avail themselves of an interplay between syntax and semantics, so I won't belabor the point.

 

[John Creighto]

Art is just a useful comparison, and ontological issues are relevent, for they are referred to in the OP itself (just look at choices #4 and #5, which directly connect mathematics to things that actually exist, i.e., claim that mathematics is explicitly ontological). I understand that you must decide what elaboration is off topic, and I might not agree with you, but it's your place to do that not mine. All I'm saying is that this whole thread is about whether mathematics is something inherently semantic and ontological (connecting with meaning in the real world, choices #4 and 5), or something inherently syntactic and epistemological (a kind of procedure for knowing that follows a fixed set of rules and has nothing to do with anything outside those rules, expressed in all the other choices). Since I am not allowed to actually support my stance, I can only state it: neither of those answers could possibly do justice to what mathematics actually is, because what mathematics actually is is a juxtaposition of those two possibilities. Mathematics takes on its meaning in the place where those things come into contact, because neither of them have any value on their own, they are both vacant notions until they are juxtaposed. To actually describe my reasoning might seem verbose, or might need to bring in how art and language also avail themselves of an interplay between syntax and semantics, so I won't belabor the point.

I tried to separate out the topic of contention into a separate thread:

Time, Ontology & Platonic Reality vs Material Reality

and I hope this will be satisfactory to the moderators. I presume that given your infraction count you will be reluctant to respond but the thread is now created in case you decide to do so. I was thinking yesterday of creating a thread of posting suggestions to help people better stay within the guidelines of the philosophy section of this forum but haven’t done So.

Peace out.

 

[audioloop]

an interesting reading about the theme...Much of modern theoretical physics assumes that the true nature of reality is mathematics. This is a great mistake. The assumption underlies most of the paradoxes of quantum mechanics, and has no empirical justification. Accepting that the assumption is wrong will allow physics and mathematics to progress as distinct disciplines...
http://fqxi.org/community/forum/top...3/__details/Schlafly_fqxischlaflynomath_1.pdf

 

[Ken G]

Yes, I agree with Schlafly. The objection is to the currently favored highly rationalistic framing of modern physics-- that it is the search for the mathematics that actually governs reality. We've made that mistake so many times in the history of physics I'm amazed we're still vulnerable to it, but it must stem from the extreme successes (not something new) that rationalistic descriptions have produced. But it seems demonstrably clear that this is just not the way mathematics is used in physics-- mathematics is part of physics, but it's not all of physics, because physics is a kind of collision between mathematics and observations, between rationalism and empiricism. Just as with the Platonism vs. arbitrariness of math that is central to this thread, or the semantics and syntax of any language (or other examples that are viewed as off topic), physics requires both rationalism and empiricism to make any sense. Neither of them by themselves is a coherent destination, and we only set ourselves up to be "shocked" yet again if we fail to remember that. So we should not attempt to decide if math is Platonic or arbitrary, or if language is semantic or syntactic, or if physics is mathematical or empirical, we should just study the interactions of all those things-- because that's what math, and language, and physics, actually are.

 

[StephenofCaly]

I tend to think that both comes from us but it seems mathematics is much more useful as a scaffolding to attach our claims about physical systems. It seems that there is something more to physical reality (or even our models of physical realty) over and above the mathematics. It seems that the mathematical theories/objects are not the same type of entities that appear to exist in the physical world. We can't get to the physical world without using mathematics because non-mathematical versions of scientific theories just seem to be practically very difficult to do. But, even though the mathematics may be indispensable and the mathematical equations we use ultimately decide what we believe about the physical world there still seems to be this difference between the two and this just adds fuel to many of the interpretative debates in science, I think.

I like this offering very much. When we discuss mathematics, we consider the mapping of an Ideal Form that exists in pure abstraction. We proceed in advancing mathematics by adding to the structure of the internal correlation in the Ideal Form of Mathematical Thought.

Mathematics has proven to be a useful tool in generating implications about the "real world," the physical world which physics studies. We once searched for the simple rules by which the real world operates, and believed that we had a finger of the laws of physics. Now, we find that we are better served by testing the consequences of a certain suggestion in Ideal Thought (Mathematics) in the physical world. This was Bohr's take on quantum mechanics, as I understand it.

Platonism (Idealism) takes the stand on the pure world of Ideal, where Forms themselves - the pure and absolute thought - are the only things that are true, and "physics" is a representation of the True. On the other hand, nominalists find that the idea of Forms is an unreal abstraction which is useful for the handling of what IS real, i.e. the physical world.

Notice how the idea of reality shifts its focus.

In any case, mathematics can be a helpful guide for how to proceed in physical inferences, and yields hypotheses readily. To date, where our concepts in physics have failed spectacularly, it is where the rules of physics which we have crafted fail to extend efficiently into particular circumstances - the very small and the very fast, for instance. We have yet to see an absolute collapse of our fundamental understandings as expressed in physical mathematics - so far, what we consider the rules of the physical world seem pretty predictable. Whether we are reading God's Mind, or just hammering together some practical observations, is the difference between Platonic Idealism and the more nominalist approaches.

"I think I think, therefore I think I am."​

 

[sigurdW]

The Philosophy of Mathemathics is essentially the Philosophy of Language!

WHY?

No Formula of Mathemathics cannot in principle be "translated" into ordinary language.
Its awkward to manage without formulae,
but there are no "pure formula" that is not a simplification of language.

 

[John Creighto]

The Philosophy of Mathemathics is essentially the Philosophy of Language!

WHY?

No Formula of Mathemathics cannot in principle be "translated" into ordinary language.
Its awkward to manage without formulae,
but there are no "pure formula" that is not a simplification of language.

This argument is like: if A is a subset of B and B is a subset of A then A=B. Yet you say math can be translated into language which I can sort of buy but when we think about language we think about non mathematical things. A cow is not a mathematical object but we can represent various aspects of a cow with mathematics. Of course you may contend that language does not really represent the cow either. Yet when we speak of a cow we at least know what we mean independently of some AI recognition system.

Math describes the structure while language describes the meaning. What is meaningful to us depends upon structure, rules and order but not all structures are meaningful to us. Perhaps an algorithm could be devised to identify which types of structures are meaningful to us or perhaps it is just a convenient mix of convention and utility.

 

[sigurdW]

This argument is like: if A is a subset of B and B is a subset of A then A=B. Yet you say math can be translated into language which I can sort of buy but when we think about language we think about non mathematical things. A cow is not a mathematical object but we can represent various aspects of a cow with mathematics. Of course you may contend that language does not really represent the cow either. Yet when we speak of a cow we at least know what we mean independently of some AI recognition system.

Math describes the structure while language describes the meaning. What is meaningful to us depends upon structure, rules and order but not all structures are meaningful to us. Perhaps an algorithm could be devised to identify which types of structures are meaningful to us or perhaps it is just a convenient mix of convention and utility.

An insightful riposte John!
Every formula is translatable into ordinary language
but not the other way round,
maths is an idealization of language.

 

[chiro]

This argument is like: if A is a subset of B and B is a subset of A then A=B. Yet you say math can be translated into language which I can sort of buy but when we think about language we think about non mathematical things. A cow is not a mathematical object but we can represent various aspects of a cow with mathematics. Of course you may contend that language does not really represent the cow either. Yet when we speak of a cow we at least know what we mean independently of some AI recognition system.

Math describes the structure while language describes the meaning. What is meaningful to us depends upon structure, rules and order but not all structures are meaningful to us. Perhaps an algorithm could be devised to identify which types of structures are meaningful to us or perhaps it is just a convenient mix of convention and utility.

You might want to think about information in terms of its role in information theory as opposed to the intuitive ideas of most people called language which is a relative and contextual thing.

Information theory defines information to have no context or interpretation at all: you have an alphabet, a collection of sentences (both finite) and a probability distribution characterizing the event space for the nature of the grammar and subsequent mappings of probability to constructed sentences.

Context basically relates pieces of information together and most language (including mathematics but it does it in a very different way to the spoken languages) is contextual and relative.

Each word that you read and the existing context creates relationships automatically in comparison to say a string of random letters which probably just confuses people.

Mathematics is actually relative and doesn't just describe structure. There are dualities everywhere in mathematics and this gives it part of its relativity. For all and there exist are dualities. The AND/OR statements in set theory are dualities. The inequalities have dualities. There are dualities within the language itself everywhere.

The dualities themselves are important because they give context to the actual descriptions just like the combination of words in a sentence give context to the other words, the entire sentence, and anything even remotely related to the ideas and terms of the sentence.

 

[sigurdW]

You might want to think about information in terms of its role in information theory as opposed to the intuitive ideas of most people called language which is a relative and contextual thing.

Information theory defines information to have no context or interpretation at all: you have an alphabet, a collection of sentences (both finite) and a probability distribution characterizing the event space for the nature of the grammar and subsequent mappings of probability to constructed sentences.

Context basically relates pieces of information together and most language (including mathematics but it does it in a very different way to the spoken languages) is contextual and relative.

Each word that you read and the existing context creates relationships automatically in comparison to say a string of random letters which probably just confuses people.

Mathematics is actually relative and doesn't just describe structure. There are dualities everywhere in mathematics and this gives it part of its relativity. For all and there exist are dualities. The AND/OR statements in set theory are dualities. The inequalities have dualities. There are dualities within the language itself everywhere.

The dualities themselves are important because they give context to the actual descriptions just like the combination of words in a sentence give context to the other words, the entire sentence, and anything even remotely related to the ideas and terms of the sentence.

Hi Chiro! You are delightfully confusing
"Duality" "Context" "information theory"?

The context of the sign is Mind and its relation to Reality
(Whatever they might be.)

 

[chiro]

Hi Chiro! You are delightfully confusing
"Duality" "Context" "information theory"
The context of the sign is Mind and its relation to Reality
(Whatever they might be.)

Duality in the above context just means a separation in the way of an inversion. A duality to a duality is just itself. In set theory, the term is "complement". If you have a universe of possibilities then the duality of A is U\A and the duality of U\A is U\(U\A) = A.

Information theory is the standard definition started mostly by Claude Shannon.

Context is just a way of saying that things are relative to one another. An example is a duality, but it is not the only form of relativity since you can have a duality relative to a subset of the universal space much like you have conditional probability that is relative to the set it is conditioned on.

In conditional probability we have P(A|B) = P(A and B)/P(B). If we let B = U (universal set) we get P(A|U) = P(A and U)/P(U) = P(A).

The duality of the event A has the probability 1 - P(A) and this has a standard interpretation in probability.

 

[sigurdW]

Excuse me my "friends". I am not all rational. I am a poem. And a formula. Fused into one.
Again:Theres pure thought. And its echo: "Rational thought"

You temporarily overload me my dear chiro. Next year? I might return? 1/Zen to you all..

 

[chiro]

Excuse me my "friends". I am not all rational. I am a poem. And a formula. Fused into one.
Again:Theres pure thought. And its echo: "Rational thought"

You will get quite a number of interpretations and they all depend on context.

A mathematician will complain that it's not logical. A linguist might make a complement regarding the creative use of grammar. An english poetry major might make some waffle comment on the prose of the piece.

A computer scientist or a logician will think of contradictions in terms of some order logic and probably think about all the paradoxes that come with logical systems.

Some people in general will think your nuts and others will think you're brilliant.

The point is that there is no unique interpretation because everyone has their own idea of what these terms mean because they all have their own context.

Also what one person calls "irrational" another finds "rational".

Usually what people try and do in all instances of trying to find an agreement when it comes to terms is that both people go back and forth until they are both satisfied coming up with something gives both parties the context that both they and the other party has.

Both party can label their own contextual definition of what something means with their own labels, but eventually the agreement will be done in a language that both can converse in, create in, and use.

But even then trying to explicitly capture context is a difficult thing because know one really knows the extent of what they know unless they get it all out explicitly. Furthermore, know one knows what others know until they do the exact same thing.

One final thing about rationality:

Think of a situation of a gambler (a problematic one: a compulsive gambler). To a statistician and a close family member of the gambler, they see what they are doing as irrational and completely without any real kind of cognitive functioning.

The gambler though has a rationale for what they are doing: they are trying to get a "return on their investment" just like a lot of people want to get a return on their investment. The investment doesn't have to be financial: it can be a time-based investment like a personal or sexual relationship or it can be a career based investment or any other "investment".

The gambler rationalizes that if they leave now, they will risk "losing" the return on their investment even though they are un-aware that the whole game is rigged so that they lose.

All actions are rationalized in some way. Whether they are "right" or "wrong" is not the main issue here: the main issue is the context and other impetus surrounding those decisions and rationalization of thoughts.

It's the same kind of mistake mathematical economists make with rational agents.

A lot of people think that being rational is maximizing your utility and doing whatever it takes to come out on top. This might be how they and their friends think, but not everyone has the same rationalization process that they do.

 

[sigurdW]

I am TRAPPED! I can't stop reading you... Youre all on my sense!
Even your errors/arrows hits target. I was leaving this poor excuse for (eh...forget it!)

Repeat ad inf.

EDIT

Youre brilliant. Your linear sharp reasoning hurts my eye!

Im retreating into a formula: Q= Reality times Reality/Ourselves

Here is a minor proof of your multi_mentality wit: "because know one really knows"

I sea it as: "because now no one really Knows" (QED)

So Y am I, a Musician, aBeing here?
I got a problem for you, lovers of truth,,,
Maths rests on PROOF! Doesnt it?

Then please prove that sentence three below does not follow,
and mind you,you are not allowed to exclude self reference!

1 Sentence 1 is not true
2 Sentence 1 = " Sentence 1 is not true "
3 Sentence 1 is true

Did I stumble upon that proof?
No, I spent thirty years in searching for it.

Aint that being silly?
Nope, we can think about ourselves,
Therefore sentences can talk about themselves...
There is no paradox...only a logical error.

 

[sigurdW]

I got a problem for you, lovers of truth,,,
Maths rests on PROOF! Doesnt it?

Then please prove that sentence three below does not follow,
and mind you,you are not allowed to exclude self reference!

1 Sentence 1 is not true
2 Sentence 1 = " Sentence 1 is not true "
3 Sentence 1 is true
.

Theres more than one proof, here's the beginning of a proof.

First we note that sentence 2 is Empirically true.
If we can show that sentence 2 is Logically false
then sentence 2 is both true and false,
And sentence 3 will not correctly follow:

2 Sentence 1 = " Sentence 1 is not true " (assumption)

We have assumed that sentence 2 is true! So...what should be done next?

 

[DragonPetter]

if you believe that mathematics arises from the properties of nature, then you would be an adherent of physism. But then how would you respond to the objection that there is so much mathematics that we do that is not directly grounded in our knowledge of the physical world?

What about mathematical equations that possesses both solutions that are grounded in the physical world and solutions that would seem impossible physically? There are solutions to equations that give positive and negative energies for example, and we simply ignore/throwaway the negative roots since our laws of nature say you can't have negative energy/mass.

It seems very convenient and illogical to throw away some of the solutions as an afterthought just because they don't fit what we observe. How can we relegate them to abstraction and blissfully use their positive root siblings as physically grounded in observation? Are the equation and its supporting mathematical structure/axioms partially grounded in the physical world to where they are only partially right in describing the universe? Or is it completely abstract, and just by coincidence some of its solutions happen to be grounded in the physical world? I have a lot of other thoughts on this that I think leads me to be a realist that views mathematics as a duality of physics (mathematics can only exist if it can be expressed in our universe, and the universe is an expression of all possible mathematics), but I'm curious if someone can resolve this. I never got a good explanation from my physics teachers other than that negative solutions "don't make sense" to reality, and I am definitely not an expert in abstract algebra theory.

With regards to the negative root example: if there is a way to compute only positive roots without acknowledging the existence of negative roots to solve a physical relationship, then I think there is more flexibility of the interpretation of mathematics. We can't even make a rule that says ignore roots < 0, because that acknowledges their existence. If we cannot escape the existence of negative roots in this computation, then we have to either give some complete explanation for them not having physical grounding or call it coincidence that the abstract physically-independent math happens to have a partial duality with physical observation, and I don't think coincidence would be a sufficient answer. Also, I'm not trying to imply that what the relationship between mathematics and the universe is hinges on this negative roots thing, that's just a common example most people can relate to.

 

[apeiron]

What about mathematical equations that possesses both solutions that are grounded in the physical world and solutions that would seem impossible physically? There are solutions to equations that give positive and negative energies for example, and we simply ignore/throwaway the negative roots since our laws of nature say you can't have negative energy/mass.

Good point. The simple answer would be that the equations describe a higher symmetry which nature then breaks. So there is more to the story than the equations can tell. Further information, further constraints, have to be supplied somehow.

 

[jackmell]

I think it's other and parton me if I don't go through all the comments to see if anyone already stated it:

Mathematics is an emergent property of a survival strategy used by life on earth: the Universe is a massively non-linear dynamo. In order to survive in such a non-linear world, life adopted a likewise non-linear dynamo: brains. The brain mimics this non-linear world and part of that mimicry is mathematics: mind, mathematics, and the Universe are of the same cloth. There would be no mathematics without this synergy between biology, evolution, selection, and the non-linear nature of the Universe.

 

[DragonPetter]

Good point. The simple answer would be that the equations describe a higher symmetry which nature then breaks. So there is more to the story than the equations can tell. Further information, further constraints, have to be supplied somehow.

I never thought about the symmetry and that is something I don't know much about, as it goes deeply into theoretical physics. It definitely does sound compelling to my layman ears tho :P

In another thread I thought of the negative roots problem as a type of computational efficiency. You have to waste some computation on abstract solutions to be able to get the physically grounded solutions. Kind of like a carnot heat engine has to lose heat (nonsense/abstract ideas) that is not usable in order to produce work that is usable (real physically grounded ideas), where both types of information, abstract and physically grounded, are represented physically (and thus everything we can imagine or discuss mathematically, abstract or not, is embedded in the universe in some way).

All of the times our brains computed the negative roots as solutions, that information was imprinted into the universe through our thought processes as configurations of memory or symbols written into paper, and has some how interacted with the universe following the laws of the universe. If a Turing machine has to first consider negative root solutions to a problem before it can reject them, then those negative roots have some physical meaning, even if they don't apply to the models that we find patterns with in the positive roots. I know that is more of an artistic/philosophical and wishful thinking relation than something that any evidence points to, but I don't know of any satisfactory or rigorous explanation for such things.

 

[apeiron]

In another thread I thought of the negative roots problem as a type of computational efficiency. You have to waste some computation on abstract solutions to be able to get the physically grounded solutions.

I see the analogy. But in the context of the OP - whether maths is generally Platonic or utilitarian - the Platonic claim would seem to be that it is not the computation that is the issue. The answers would "exist" regardless of whether some human calculated them.

The difficulty comes only at the point of chosing one computation to be real, and discarding the other one as unphysical. And this seems a non-mathematical decision. The maths itself does not offer the grounds for making the choice.

All of the times our brains computed the negative roots as solutions, that information was imprinted into the universe through our thought processes as configurations of memory or symbols written into paper, and has some how interacted with the universe following the laws of the universe.

But how is there any interaction apart from that we create ourselves? This is where the utilitarian part of the story comes in. To the universe, any symbols scribbled out on paper are just noise - meaningless entropy. It is only to our minds that the symbols are information - one root being the definitely true, the other being the definitely false. And our minds decide this through further measurement - we observe the world and make the distinction.

 

[DragonPetter]

I see the analogy. But in the context of the OP - whether maths is generally Platonic or utilitarian - the Platonic claim would seem to be that it is not the computation that is the issue. The answers would "exist" regardless of whether some human calculated them.

Well, I only use human computation as the example. I am talking purely computational in the generic sense, like a Turing machine. A computer has no mind or existence, it is an object, and yet it can compute math based on what the laws of the universe allow it do. It might take infinitely long for it to reach some results, but it is still only able to get results that the laws of the universe let it reach.

I suppose it might still be debatable that if everything a human mind can do, a Turing machine can do too, but if that is true then anything our minds think up has to be processed in adherence to the laws of physics just like for a Turing machine. If you write a computer program to find the roots of a solution to give you an answer that matches physical reality, the throw away roots are still processed. At some point we make a decision to stop finding roots or throw away roots, which costs energy and creates heat by the computer, and so I think it might be more than just an analogy.

The difficulty comes only at the point of chosing one computation to be real, and discarding the other one as unphysical. And this seems a non-mathematical decision. The maths itself does not offer the grounds for making the choice.

Yes, I agree to a certain degree. I am taking the step to say that the universe's rules makes the choice automatically, but the universe's rules also allowed for the existence of the negative root solution as well, even if it has no other physical grounding to the world than its own existence. Perhaps, by the universe's rules, you cannot generate some solutions without creating an abstract "waste" solution. That's why I am confused/annoyed/interested that we think we can make such non-mathematical decisions about a purely mathematical result. We break away from the math at the last minute, even though part of its result is what we want, and then we go on happily using the math that works (ya, I'm trying to stress that analogy) for us. But if what you suggested is the case, that the universe breaks symmetry from math, then the universe is not purely mathematical or does not include all math. If that is the case, I would guess that we should not even be aware of these abstract ideas.

But how is there any interaction apart from that we create ourselves? This is where the utilitarian part of the story comes in. To the universe, any symbols scribbled out on paper are just noise - meaningless entropy. It is only to our minds that the symbols are information - one root being the definitely true, the other being the definitely false. And our minds decide this through further measurement - we observe the world and make the distinction.

To the first question: Our brains are part of the universe, and so the interaction is with parts of the universe. The neurons exchange signals, consume energy, generate heat, organize synapses to form ideas, etc. This is all physically governed by the laws of the universe. If the laws of the universe don't allow our brains to do something, then there is no possibility for it to exist in our thoughts. Likewise, if our brains can process the information that generates abstract mathematical logic, then its only because it is built into the interaction of the universe's laws.

Also, I don't see the organization of those symbols as noise if there is no brain around to interpret them. That organization of symbols still exists, regardless of anyone to interpret it. I don't know for sure though :P

 

[Pythagorean]

From a science perspective, when doing mathematics, really getting into a problem... pages of derivation, especially based in physics... it's really easy to slip into a platonist mind set. Perhaps even advantageous.

When you actually go and try to apply models to the real world and do experiments, and raise all the caveats that come from a complex and rich world, physism/formalism emerges.

 

[apeiron]

If you write a computer program to find the roots of a solution to give you an answer that matches physical reality, the throw away roots are still processed. At some point we make a decision to stop finding roots or throw away roots, which costs energy and creates heat by the computer, and so I think it might be more than just an analogy.

Yes, all actual computation will generate entropy (putting aside http://en.wikipedia.org/wiki/Landauer's_principle for the moment). But the central trick of computation is precisely that it minimises any interaction with the world - so as to put itself into a "Platonic" realm.

A sequential symbol processing computer - a Turing machine - executes any individual step with exactly the same heat dissipation. So as far as the world knows (as far as the second law cares), calculating nonsense looks the same as calulating mentally-significant results. Inside the machine, spitting out a positive or negative answer is still a symmetric situation as the entropic cost is precisely the same.

The real impact on the world only comes from the actions people take based on what they believe. Some further choice has to be made as to which answer is the correct one. Further energy is required to break the entropic symmetry of the computational result, even if some entropic effort was required to produce that result - get things to the stage of a choice between a positive and negative root in your example.

I am taking the step to say that the universe's rules makes the choice automatically, but the universe's rules also allowed for the existence of the negative root solution as well, even if it has no other physical grounding to the world than its own existence.

But the laws of physics are a human invention. They may certainly encode some regularity, some generality, that describes nature. But it is falling back into the confusion of Platonism to mistake our models of reality with reality itself.

So here it is our model (expressed in mathematical statements) that allows for a symmetric pair of choices. The Universe just does what it does and if our model can't predict that, then this is just a sign of its incompleteness.

It is the model that has rules. And it is the rules themselves which create the appearance of choice. The Universe by contrast exists in time and has locked in its critical "choices". (See http://en.wikipedia.org/wiki/Loschmidt's_paradox).

But if what you suggested is the case, that the universe breaks symmetry from math, then the universe is not purely mathematical or does not include all math. If that is the case, I would guess that we should not even be aware of these abstract ideas.

No, my argument is that we create math (or rather, our mathematical descriptions of material reality) by stepping back from the current broken symmetry we see all around to recover the original symmetry that must have been the Universe's initial state.

We are modellers, so there is no problem with being aware of our own created abstractions.

To the first question: Our brains are part of the universe, and so the interaction is with parts of the universe. The neurons exchange signals, consume energy, generate heat, organize synapses to form ideas, etc. This is all physically governed by the laws of the universe. If the laws of the universe don't allow our brains to do something, then there is no possibility for it to exist in our thoughts. Likewise, if our brains can process the information that generates abstract mathematical logic, then its only because it is built into the interaction of the universe's laws.

Humans cannot ultimately escape the second law of thermodynamics (the relevant law here). But again, the whole point about computation (and modelling in general) is that it allows for the kind of temporary escape available to life/mind as an order-creating dissipative structure - http://merkury.orconhosting.net.nz/lifeas.pdf

So computation must create heat in practice. But it is useful because it demands so little energy compared to the amount of energy it allows us to harness. And the energy consumption is the same regardless of whether we are computing sense or nonsense - which is what gives us free choice about what to compute, what results to generate.

So the second law can't effectively see what we are doing inside our heads, or computing inside our computers. We have created a private Platonic realm of pure thought and choice. On the larger scale of course, the second law does rule. We have to eat to think, plug in our computers to compute. But that still leaves us a fantastic amount of Platonic freedom to play around in.

Also, I don't see the organization of those symbols as noise if there is no brain around to interpret them. That organization of symbols still exists, regardless of anyone to interpret it. I don't know for sure though :P

This is the symbol grounding problem. And quite clearly symbols require interpreters.

Information theory can be used to model an observerless reality. But that is just another of our useful modern abstractions that should not be mistaken as the deep truth of reality.

Bits are just entropy - countable states. To be "orderly", they have to also be placed within an interpretive context. Someone has to care enough to count.

 

[Noja888]

If we could measure all components of a human mind can we predict what choices will be made by that mind? - Or - I like to think we end up at such small scales that quantum mechanics plays a role. The turing machine becomes the incorrect anology perhaps. A quantum turing machine, where the outcome always has some degree of uncertaincy. The act of measurement disturbs the system, our freewill is a manifestation of this concept in someway perhaps. By studying someones likes and dislikes you can reasonabliy predict how they will act in a particular situation. But until they make their choice you cannot say for certain the outcome. To me atleast freewill seems linked to the uncertaincy princeable in quantum mechanics. has anyone heard of someone theorise a strong AI system with quantum computing? IBM seems a few years off from creating a quantum computer. Would be great to see if that is possible. On a sidenote I believe any mathematical construct has an application because it was created in this universe

 

[chiro]

Noja888: You might want to consider whether the information is even local or not: what are you going to do when it isn't local?

As an interesting thing to ponder: consider the idea of the "soul".

A lot of people when they tried to find the soul, cut up the body and were looking for this thing called the soul and unsurprisingly, no one could find a single element that they called the soul.

You might laugh at this, but in a large sense, we still analyze in this exact same way: we isolate and divide in a way where we segregate things into mutually exclusive parts and consider that the totality of the system and its context is within some sort of isolated boundary.

But how are you going to even find the information if most of it doesn't even exist within the boundary?

 

[Evo]

This thread should have been locked a LONG time ago.