/ Is mathematics invented or discovered?

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Jimbo W on 16 Jan 2013
So following on from the prechristmas discussion of dcistence/being and what can and can't be said about it, i have been wondering about mathematics, is it something we discover, can it evolve or do we invent it?
If we discover it, does it exist apart from the presence of an agency to perceive it?
If it is invented, why does it appear so immutable?
Is it a static thing, or does it evolve e.g. Vis a vis chaos and fractals?
Why does it appear so useful in the scientific abstraction of reality?
What conceivable reason is there for us to be able to comprehend / discover differential equations, from an human evolutionary point of view?
Is mathematics what lies at the heart of all reality as Max Tegmark suggests?
Timmd on 16 Jan 2013
In reply to Jimbo W:
> So following on from the prechristmas discussion of dcistence/being and what can and can't be said about it, i have been wondering about mathematics, is it something we discover, can it evolve or do we invent it?

> If we discover it, does it exist apart from the presence of an agency to perceive it?

I think so, yes.

> If it is invented, why does it appear so immutable?



> Is it a static thing, or does it evolve e.g. Vis a vis chaos and fractals?

I think we creat different ways of thinking, but the formulae behiend how the world/universe is shaped are just waiting to be stumbled upon.

> Why does it appear so useful in the scientific abstraction of reality?

Because everything follows mathematical patterns, I think.

> What conceivable reason is there for us to be able to comprehend / discover differential equations, from an human evolutionary point of view?

To be able to understand the world more fully.

> Is mathematics what lies at the heart of all reality as Max Tegmark suggests?

I think so.
crossdressingrodney - on 16 Jan 2013
In reply to Jimbo W:
Interesting questions.

My feeling is that when you prove a new theorem, you have not created something, but rather uncovered what was already there, waiting to be found. The statement and proof of the theorem surely existed before you happened to write them down. It seems to me that mathematics is a static, immutable thing.

> or does it evolve e.g. Vis a vis chaos and fractals?

What do you mean by this? In what way do chaos or fractal evolve?

> Is mathematics what lies at the heart of all reality as Max Tegmark suggests?

Certainly seems that way to me. The mathematics of everyday experience can be dull and messy, but as you probe more and more fundamental physics, it is described by ever deeper and more wonderful mathematics.
Rob Exile Ward on 16 Jan 2013
In reply to Jimbo W: Dunno.
Robert Durran - on 16 Jan 2013
In reply to Jimbo W:
> If we discover it, does it exist apart from the presence of an agency to perceive it?

I suspect so. A prime will still be a prime number when all life in the universe has died out.

> Is it a static thing, or does it evolve e.g. Vis a vis chaos and fractals?

Not sure what you mean by that.

> Why does it appear so useful in the scientific abstraction of reality?

Maybe we tend to study the subset of mathematics which is motivated by or describes reality. Maybe reality is fundamentally mathematical. Or maybe reality just does its "own thing" and we do our best to describe it with mathematical laws which have no really deep relation to reality (though I hope not).

> What conceivable reason is there for us to be able to comprehend / discover differential equations, from an human evolutionary point of view?

As an accidental byproduct of other cognitive abilities which we evolved in order to survive on the african Savannah? Some would say of language.

> Is mathematics what lies at the heart of all reality as Max Tegmark suggests?

Maybe. Or maybe it is actually external to reality.

Offwidth - on 16 Jan 2013
In reply to crossdressingrodney:

"> Is mathematics what lies at the heart of all reality as Max Tegmark suggests?

Certainly seems that way to me. The mathematics of everyday experience can be dull and messy, but as you probe more and more fundamental physics, it is described by ever deeper and more wonderful mathematics."

A remarkably common view amongst scientists but oddly unscientific (we should all be keeping an open sceptical mindset)
dissonance - on 16 Jan 2013
In reply to Jimbo W:
> So following on from the prechristmas discussion of dcistence/being and what can and can't be said about it, i have been wondering about mathematics, is it something we discover, can it evolve or do we invent it?

under patent law its something discovered eg you cant patent it.

> What conceivable reason is there for us to be able to comprehend / discover differential equations, from an human evolutionary point of view?

there is unlikely to be one directly. However the reasoning methods etc come in useful in general.

Robert Durran - on 16 Jan 2013
In reply to Offwidth:
> (In reply to crossdressingrodney)
> Certainly seems that way to me. The mathematics of everyday experience can be dull and messy, but as you probe more and more fundamental physics, it is described by ever deeper and more wonderful mathematics."
>
> A remarkably common view amongst scientists but oddly unscientific (we should all be keeping an open sceptical mindset)

But it's not really scientific thing that can be tested scientifically any more than whether the Mona Lisa is beautiful.

The beauty of mathematics is a subjective thing, but there seems no denying that there is a certain consensus that the mathematics we use to describe reality becomes more beautiful at more fundamental level of physics (though there is obviously great beauty at higher levels too - chaos and complexity for example). Perhaps, in a sense, "simple" would be a better word in that the fundamental laws can be expressed very concisely at the deeper levels.

MG - on 16 Jan 2013
In reply to dissonance:

> there is unlikely to be one directly. However the reasoning methods etc come in useful in general.


Are you sure? Activities such as catching things all require the solutions to DEs, even if we solve them rapidly in our heads rather than writing things down formally.
Robert Durran - on 16 Jan 2013
In reply to MG:
> (In reply to dissonance)
> Activities such as catching things all require the solutions to DEs, even if we solve them rapidly in our heads rather than writing things down formally.

Really? Surely it can be done just by experience and memory of how balls move through air under gravity of 9.8m/ss. I bet you'd have trouble catching balls on the moon until you got used to lower gravity.

Coel Hellier - on 16 Jan 2013
In reply to Jimbo W:

> Is mathematics what lies at the heart of all reality as Max Tegmark suggests?

Mathematics is the study of patterns. And it does seem that, at root, reality is patterns, patterns of matter. These patterns are determined by the properties of particles, how they interact with each other. Or, putting that another way, the properties of a particle are how it interacts with other things, and those interactions give rise to patterns, and the study of those patterns is mathematics.

Thus mathematics is an abstraction about reality, a distillation of its fundamental properties. Thus maths is discovered, indeed it is derived empirically, derived from the study of the universe.

> If we discover it, does it exist apart from the presence of an agency to perceive it?

Yes (provided that we accept that reality exists outwith our human ability to perceive it).

> Why does it appear so useful in the scientific abstraction of reality?

Because mathematics *is* the scientific abstraction of reality.

> What conceivable reason is there for us to be able to comprehend / discover differential equations,
> from an human evolutionary point of view?

Understanding the reality around us is highly beneficial from an human evolutionary viewpoint. Evolution thus programs a general-purpose thinking machine, which can then turn its attention to matters that go beyond evolutionary utility.
Edradour - on 16 Jan 2013
In reply to Robert Durran:
> (In reply to Jimbo W)
> [...]
>
> I suspect so. A prime will still be a prime number when all life in the universe has died out.
>

But a prime number is a concept that only exists to humans. In fact a number is only a concept that exists to humans... Therefore, they won't exist after all life has died out in the same way that the colour blue won't. It may still be there but it won't necessarily be described as 'blue'.

Maths is another form of language in my opinion and we use it to explain the world. The world doesn't revolve around maths because only humans are aware of its existence.

Is a beautiful book or poem invented or discovered?

Coel Hellier - on 16 Jan 2013
In reply to Fickalli:

> But a prime number is a concept that only exists to humans. In fact a number is only
> a concept that exists to humans.

That's dangerous talk, using the word "exists" like that -- we could be here all month! ;-)
crossdressingrodney - on 16 Jan 2013
In reply to Offwidth:
> > > Is mathematics what lies at the heart of all reality as Max Tegmark suggests?

> > Certainly seems that way to me. The mathematics of everyday experience can be dull and messy, but as you probe more and more fundamental physics, it is described by ever deeper and more wonderful mathematics."

> A remarkably common view amongst scientists but oddly unscientific (we should all be keeping an open sceptical mindset)

Not sure what you mean. It's neither unscientific nor closed-minded to draw conclusions based on experience.

What exactly are scientists not sceptical enough about and what should they do about it? Spend more time trying to describe reality without maths?
999thAndy on 16 Jan 2013
In reply to Fickalli:
> (In reply to Robert Durran)
> [...]
>
> But a prime number is a concept that only exists to humans. In fact a number is only a concept that exists to humans... Therefore, they won't exist after all life has died out in the same way that the colour blue won't. It may still be there but it won't necessarily be described as 'blue'.
>

Regardless of the name, the colour or the number will be there long after we've died

http://www.bbc.co.uk/news/magazine-14305667
MG - on 16 Jan 2013
In reply to Robert Durran:
> (In reply to MG)
> [...]
>
> Really? Surely it can be done just by experience and memory of how balls move through air under gravity of 9.8m/ss. I bet you'd have trouble catching balls on the moon until you got used to lower gravity.

I don't know. You can catch a ball on a windy day, for instance, when its trajectory will be quite different to a still day. And isn't memory just an approach to solving the equations?

Coel Hellier - on 16 Jan 2013
In reply to Robert Durran:

>> Activities such as catching things all require the solutions to DEs, ...

> Really? Surely it can be done just by experience and memory of how balls move through air under gravity of 9.8m/ss.

I think you're saying the same thing. If you define "solving" the DE to be the ability to predict the location of the ball at different times, then both the "experience" way and the maths way count as "solving" the DE.
alanw - on 16 Jan 2013
In reply to Jimbo W: Interesting questions indeed that could take a lifetime to answer. Like most replies so far I would agree that at it's core maths is discovered but I do think there is an element of invention as well.

If you take a fairly basic concept like numbers I'd say that the concept of positive whole numbers is fundamental to the universe. But how we represent them can be very different. Compare, for example the numbers we use today with those the Romans used. Both of these are rather different to the way a hard core number theorist conceives of numbers.

Proofs also vary. I'd say Pythagoras' theorem is a fundamental property of flat space that is discovered but there are numerous different ways to prove it; are each of these proofs discovered or invented?

I've often thought that this would be one question that would benefit greatly from discovering alien civilisations. I suspect that other advanced species would have a lot of maths in common with us but possibly very different ways of working things out. The common stuff would be the discovered bits and the different methods the invented stuff.

As to why maths is so good at describing the world, I just get a headache when I think too much about that.
Edradour - on 16 Jan 2013
In reply to 999thAndy:
> (In reply to Fickalli)
> [...]
>
> Regardless of the name, the colour or the number will be there long after we've died
>
> http://www.bbc.co.uk/news/magazine-14305667

Not quite my point. Maths and numbers have been developed by humans to describe patterns in the world around us. Only humans are aware of the concept. Therefore, whilst the pattern of 'prime numbers' will be there after we are long gone 'prime numbers' will not because they are a human construct. Similarly the word 'blue' has been developed to describe a particular process that happens in the eye and brain. That process may exist after humans but 'blue' will not.

ablackett - on 16 Jan 2013
In reply to Fickalli:
> (In reply to Robert Durran)
> [...]
>
> But a prime number is a concept that only exists to humans.

That's not true. Cicadas insects in north america have a pretty good understanding of them.

http://www.bbc.co.uk/news/magazine-14305667

And, although we can't prove it, i'm pretty sure that any alien life forms would also understand primes if they were intelligent enough to contact us.

One of the first theorems that I was taught at university was called the Fundamental Theorem of Arithmetic, which states that every integer greater than 1 can be expressed as a unique product of primes. It is probably as important to mathematics as the fact that all matter is made up of elements is important to chemists, both things were discovered, not invented.
crossdressingrodney - on 16 Jan 2013
In reply to Fickalli:
> Maths is another form of language in my opinion and we use it to explain the world.

Maths is a language, but it's not the same type of thing as, say, the English language. Maths is a language derived naturally from how the universe is. For example, intelligent life in a distant galaxy would also know about prime numbers. Obviously, they would not speak English because English is just an accident of history.

> The world doesn't revolve around maths because only humans are aware of its existence.

I'd say the world does revolve around maths, and your argument doesn't hold because any intelligent life could become aware of maths, not just humans. After all, babies very quickly develop an intuitive understanding of the natural numbers; in fact, I'd be surprised if various mammals didn't have an appreciation of the notion of natural numbers.
dissonance - on 16 Jan 2013
In reply to MG:

> Are you sure? Activities such as catching things all require the solutions to DEs, even if we solve them rapidly in our heads rather than writing things down formally.

in terms of solving them on paper, yes. turning it into a deliberate concious item is something else.
winhill - on 16 Jan 2013
In reply to Fickalli:
> (In reply to 999thAndy)
>
> Not quite my point. Maths and numbers have been developed by humans to describe patterns in the world around us. Only humans are aware of the concept. Therefore, whilst the pattern of 'prime numbers' will be there after we are long gone 'prime numbers' will not because they are a human construct. Similarly the word 'blue' has been developed to describe a particular process that happens in the eye and brain. That process may exist after humans but 'blue' will not.

But this is the same as saying there is a difference between an English speaker talking about numbers or a French speaker talking about numbers, which there isn't (apart from trivial sounds).
Edradour - on 16 Jan 2013
In reply to ablackett:
> (In reply to Fickalli)
> [...]
>
> That's not true. Cicadas insects in north america have a pretty good understanding of them.
>
> http://www.bbc.co.uk/news/magazine-14305667
>
> And, although we can't prove it, i'm pretty sure that any alien life forms would also understand primes if they were intelligent enough to contact us.
>

Yeah, I read it the first time it was linked to in reply to my post! :)

Like I said, the pattern will still exist but the cicada insect isn't sitting there thinking '1,3,5...13, excellent I'll come out now' because that language, both numbers and words are a human construct.

999thAndy on 16 Jan 2013
In reply to Fickalli:

Maths was described to me as islands of knowledge in a sea of ignorance, and new maths builds bridges or extends those islands. If we all die tomorrow then yes the sea of ignorance sweeps back in. However when new life evolves to the point where it can create maths, they will rediscover the same islands.
Edradour - on 16 Jan 2013
In reply to crossdressingrodney:
> (In reply to Fickalli)
> [...]
>
> I'd say the world does revolve around maths, and your argument doesn't hold because any intelligent life could become aware of maths, not just humans.

Really? They could become aware of the patterns in the natural world but might describe them in a different way which would then be a completely different version of 'maths'.

> After all, babies very quickly develop an intuitive understanding of the natural numbers; in fact, I'd be surprised if various mammals didn't have an appreciation of the notion of natural numbers.

What do you mean by natural numbers?
Edradour - on 16 Jan 2013
In reply to 999thAndy:
> (In reply to Fickalli)
>
> Maths was described to me as islands of knowledge in a sea of ignorance, and new maths builds bridges or extends those islands. If we all die tomorrow then yes the sea of ignorance sweeps back in. However when new life evolves to the point where it can create maths, they will rediscover the same islands.

Nice. I like that.

999thAndy on 16 Jan 2013
In reply to Fickalli: You're welcome
victim of mathematics - on 16 Jan 2013
In reply to Fickalli:
> (In reply to crossdressingrodney)
> [...]
>
> Really? They could become aware of the patterns in the natural world but might describe them in a different way which would then be a completely different version of 'maths'.

No. The notation might be different, but the underlying maths would be the same.

>
> [...]
>
> What do you mean by natural numbers?

1,2,3,4...

Scarab9 - on 16 Jan 2013
In reply to Jimbo W: #

the solutions we find to explain how things work are discovered.

the methods used to find the solutions are invented (or developed).

It depends on whether you're referring to mathematics as in the building blocks of nature or as the scientific practice.
ads.ukclimbing.com
Robert Durran - on 16 Jan 2013
In reply to MG:
> (In reply to Robert Durran)
>
> I don't know. You can catch a ball on a windy day, for instance, when its trajectory will be quite different to a still day.

Presumably by continuously updating it's estimated position and velocity and adjusting your own position accordingly. Harder than on a windless day though!

Edradour - on 16 Jan 2013
In reply to victim of mathematics:
> (In reply to Fickalli)
> [...]
>
> No. The notation might be different, but the underlying maths would be the same.
>

Is there not an element of hubris in that attitude? That the human interpretation of the patterns in the natural world is absolutely the correct one?

Throughout history we, as a species, have believed things that have subsequently proven not to be true; the world is flat, the sun orbits the earth, the atom is the smallest particle etc. Is there not a chance that there could be an alternative, equally as valid, interpretation of the natural world that isn't 'maths' as we perceive it?

Robert Durran - on 16 Jan 2013
In reply to Coel Hellier:
> (In reply to Robert Durran)
> I think you're saying the same thing. If you define "solving" the DE to be the ability to predict the location of the ball at different times, then both the "experience" way and the maths way count as "solving" the DE.

Yes, but I presume the OP meant the maths way!

Robert Durran - on 16 Jan 2013
In reply to ablackett:
> (In reply to Fickalli)
> That's not true. Cicadas insects in north america have a pretty good understanding of them <prime numbers>

No they don't. Evolution has just stumbled upon one or two of them by chance.
Robert Durran - on 16 Jan 2013
In reply to Fickalli:
> But a prime number is a concept that only exists to humans.

So are you seriously saying that 13 might cease to be a prime number and suddenly acquire factors if human life (or indeed all life) is wiped out. Given that numbers of things will still exist, that really does seem an absurd idea.
Edradour - on 16 Jan 2013
In reply to Robert Durran:
> (In reply to Fickalli)
> [...]
>
> So are you seriously saying that 13 might cease to be a prime number and suddenly acquire factors if human life (or indeed all life) is wiped out. Given that numbers of things will still exist, that really does seem an absurd idea.

No, I'm saying that '13', 'prime number' and 'factors' and 'numbers' are human constructs to describe things that exist naturally.

Robert Durran - on 16 Jan 2013
In reply to Coel Hellier:
>
> Mathematics is the study of patterns. And it does seem that, at root, reality is patterns, patterns of matter. These patterns are determined by the properties of particles, how they interact with each other. Or, putting that another way, the properties of a particle are how it interacts with other things, and those interactions give rise to patterns, and the study of those patterns is mathematics.
>
> Thus mathematics is an abstraction about reality, a distillation of its fundamental properties. Thus maths is discovered, indeed it is derived empirically, derived from the study of the universe.

But don't you think we could come up with some really wacky axioms and develop some mathematics which doesn't in any way relate to external reality? Or are all possible patterns out there somewhere in reality? Or do you consider the wacky axioms and weird ensuing mathematics fundamentally part of reality? Or alternatively external to reality?
Robert Durran - on 16 Jan 2013
In reply to Fickalli:
> (In reply to Robert Durran)
>
> No, I'm saying that '13', 'prime number' and 'factors' and 'numbers' are human constructs......

So you are just playing with words! Thirteenness is still thirteenness whether you call it "thirteen" or "treize".

> ........to describe things that exist naturally.

ie we are basically in agreement!



victim of mathematics - on 16 Jan 2013
In reply to Fickalli:
> (In reply to victim of mathematics)
> [...]
>
> Is there not an element of hubris in that attitude? That the human interpretation of the patterns in the natural world is absolutely the correct one?
>
> Throughout history we, as a species, have believed things that have subsequently proven not to be true; the world is flat, the sun orbits the earth, the atom is the smallest particle etc. Is there not a chance that there could be an alternative, equally as valid, interpretation of the natural world that isn't 'maths' as we perceive it?

I think you have a different perception of maths than I do. A prime number IS a prime number, there's no interpretational element to it.

There's an important difference between the methods of broader science (thesis, antithesis, synthesis), which can be disproved, and those of maths (axiomatic, logical) which cannot be disproved (although somebody might make a mistake somewhere along the line or you might object to their axioms). Perhaps that's where the confusion arises?
1poundSOCKS - on 16 Jan 2013
In reply to Robert Durran: I suppose Newton's theory of gravity is technically wrong, but it's still a useful tool, and maybe the same can be said about relativity because it falls down in certain extreme conditions.
Jon Stewart - on 16 Jan 2013
In reply to Fickalli:
> (In reply to victim of mathematics)
> [...]
>
> Is there not an element of hubris in that attitude? That the human interpretation of the patterns in the natural world is absolutely the correct one?

No, because our interpretation of the patterns is continually being used to make useful predictions of how nature will behave. Our interpretation of nature remains anchored to the reality of nature through this process - experiments. If we're wrong, we revise the interpretation making sure we don't go off on some silly tangent.

> Throughout history we, as a species, have believed things that have subsequently proven not to be true; the world is flat, the sun orbits the earth, the atom is the smallest particle etc. Is there not a chance that there could be an alternative, equally as valid, interpretation of the natural world that isn't 'maths' as we perceive it?

When we revise a theory, we don't just change our minds from one thing which is untrue to another thing which could be equally untrue. Take for example the world being flat. It is, locally. The later description had to take into account the fact that a flat map represents the local area rather well, and does so by saying that the ball we live on is big enough not to notice the curvature all the time. Each new theory has to account for why the old one gave the observations it did, as well as the anomalies that had been noticed (like the horizon that you could never seem to reach).

So we know that our theories are descriptions of reality, and an alien species doing its own analysis of the patterns of nature might find a different way to express its understanding of the patterns. But it would be the same maths in a different notation because it lives in the same universe with the same patterns.

[So yes, maths is discovered, not invented. I was a bit disappointed not to see a bunch of wishy-washy people arguing that everything is a social construct, who could then be shot down and have their whole view of the universe reconstructed into something that isn't complete bullshit.]
Edradour - on 16 Jan 2013
In reply to Jon Stewart:
> (In reply to Fickalli)
> [...]
>
>
> [So yes, maths is discovered, not invented. I was a bit disappointed not to see a bunch of wishy-washy people arguing that everything is a social construct, who could then be shot down and have their whole view of the universe reconstructed into something that isn't complete bullshit.]

Is that a reference to me?

I'm a structural engineer by training, use maths everyday and am actually just playing devils advocate but an interesting thread nonetheless....

Jon Stewart - on 16 Jan 2013
In reply to Fickalli:

No, take it at face value!
Jon Stewart - on 16 Jan 2013
In reply to 1poundSOCKS:
> (In reply to Robert Durran) I suppose Newton's theory of gravity is technically wrong, but it's still a useful tool, and maybe the same can be said about relativity because it falls down in certain extreme conditions.

I don't think it's wrong. It's just a bit less right than General Relativity. Which in turn might be a bit less right than the next great theory of gravity. They're all just descriptions of the patterns as they appear at different levels of scrutiny.

I find this feature of the way nature works amazing. Why should the bizarre pattern described by General Relativity have a convenient approximation that can be worked out in 17th century 'on the way' to the deeper pattern? Why should electromagnetism follow beautiful patterns described by classical physics when it's a much more weird phenomenon at heart?
cb294 - on 16 Jan 2013
In reply to Robert Durran:
I would argue that the core of mathematics lies in its interal consistency. I am sure it is possible to generate formal rule sets for manipulating variables (mathematics) that are internally consistent but do not describe the behaviour of things as observed in reality.

We have selected what we call mathematics in everyday terms from a set of different conceivable mathematics because it does describe our reality and is thus useful, but probably not really necessary.

Could be wrong, though....

CB
1poundSOCKS - on 16 Jan 2013
In reply to Jon Stewart: Lol, I knew I shouldn't post on a science thread! :) I don't know, maybe I could come up with an even less general theory than Newtonian gravity that predicts incorrectly most of the time, and occasionally it get quite close to observation. Was that invented by me, or discovered?
Jon Stewart - on 16 Jan 2013
In reply to 1poundSOCKS:
> (In reply to Jon Stewart) I don't know, maybe I could come up with an even less general theory than Newtonian gravity that predicts incorrectly most of the time, and occasionally it get quite close to observation. Was that invented by me, or discovered?

Great idea. While you're working on that, I'll try for a theory of how the different species came about which is a sort of half-way house between Adam and Eve and Darwin. Say, god creating the animals so he had something to hunt for sport, and to make it more fun, he gave them the ability to get better at surviving, so as his skill increased the game got harder. I like this...

Coel Hellier - on 16 Jan 2013
In reply to Robert Durran:

> But don't you think we could come up with some really wacky axioms and develop some mathematics
> which doesn't in any way relate to external reality?

I guess we can conceive (sort of) of different realities that run on totally alien logic, that is incompatible with the logic of our universe.

> Or are all possible patterns out there somewhere in reality?

Not in our "reality". Other maths/logic systems could perhaps "exist" in other "meta-realities".
Robert Durran - on 16 Jan 2013
In reply to 1poundSOCKS:
> (In reply to Jon Stewart)I don't know, maybe I could come up with an even less general theory than Newtonian gravity that predicts incorrectly most of the time.

You certainly could. Flat earth low altitude gravitation: everything has a downward force of mg on it. Does fine for most everyday things.
Robert Durran - on 16 Jan 2013
In reply to Coel Hellier:
> (In reply to Robert Durran)
> I guess we can conceive (sort of) of different realities that run on totally alien logic, that is incompatible with the logic of our universe.

I'm not sure we need different logic. Just different axioms. The axioms of euclidean geometry don't require different logic from non Euclidean geometry do they?
1poundSOCKS - on 16 Jan 2013
In reply to Jon Stewart: Not sure I can come up with a new theory of gravity quite so quickly, flat earth or not!
Robert Durran - on 16 Jan 2013
In reply to Robert Durran:
> (In reply to 1poundSOCKS)
> You certainly could. Flat earth low altitude gravitation: everything has a downward force of mg on it. Does fine for most everyday things.

Just like Newtonian Gravity does fine for sending rockets to the moon etc.

Jimbo W on 16 Jan 2013
In reply to dissonance:

> under patent law its something discovered eg you cant patent it.

Unbelievably, some scientists have successfully patented newly discovered genes!
1poundSOCKS - on 16 Jan 2013
In reply to Robert Durran: So primitive humans 'discover' flat earth gravity, then find out the earth isn't flat...where does that leave us?
Robert Durran - on 16 Jan 2013
In reply to 1poundSOCKS:
> (In reply to Robert Durran) So primitive humans 'discover' flat earth gravity, then find out the earth isn't flat...where does that leave us?

We modify it to get Newtonian Gravity.

And later modify that to get General Relativity.

And hopefully later modify that to get Quantum Gravity.

Jimbo W on 16 Jan 2013
In reply to Coel Hellier:

> Mathematics is the study of patterns. And it does seem that, at root, reality is patterns, patterns of matter. These patterns are determined by the properties of particles, how they interact with each other. Or, putting that another way, the properties of a particle are how it interacts with other things, and those interactions give rise to patterns, and the study of those patterns is mathematics.
> Thus mathematics is an abstraction about reality, a distillation of its fundamental properties. Thus maths is discovered, indeed it is derived empirically, derived from the study of the universe.

So in that case, is mathematics provisional in the same way that scientific theories about reality are provisional? If not, why not - how can they be appreciable properties discovered as a facet of material behaviour and yet be apparently so much less mutable? If, rather, mathematics is provisional, in what way is it provisional (could Pi or basic geometry vary with different dimensional perspectives, i.e. do these basic mathematical properties depend on a relative dimensional point of view or vary along with the environmental physical laws that may emerge in different universes?), or are you refering to the intrinsic incompleteness of axiomatic systems discussed by Godel?

> Yes (provided that we accept that reality exists outwith our human ability to perceive it).

So reality and maths are both agency independent?

> Because mathematics *is* the scientific abstraction of reality.

Is it? Or is it a tool that we use to access it? Is it more fundamental than that?

> Understanding the reality around us is highly beneficial from an human evolutionary viewpoint. Evolution thus programs a general-purpose thinking machine, which can then turn its attention to matters that go beyond evolutionary utility.

Why the machine metaphor, and why the teleological anthropomorphic reasoning? And while it appears obvious that organisms benefit from a comprehension of small natural numbers, why does that sort of abstractive ability infer much more complex abstractions? What sort of environmental evolutionary pressure would drive this?
1poundSOCKS - on 16 Jan 2013
In reply to Robert Durran: Ha ha, yes. I was thinking about the original questions, invented vs. discovered.
shaymarriott - on 16 Jan 2013
In reply to Jimbo W:

Advanced apologies if I'm repeating points already made, I haven't time to read all the posts... but here are my 2 cents...

I don't think mathematics can be looked at in terms of being 'discovered', as surely it constitutes the most raw, basic essence of the way the universe and everything within it functions. Humans have 'discovered' maths by attempting to understand more and more complex patterns and theories, but the maths was always there. Think about Gravity - sure, Newton came up with some interesting insights into what we now know about it (and some might try and argue at a basic level that he discovered it) but it was undeniably always there. It has always been experienced as an integral part of our existence, but no-one knew what it was.

'Mathematics' (as a descriptive term) exists purely as our perception of the rules of the universe, and incorporates physics amongst other things. What mathematics describes and attempts to understand has always existed, and is far more complex than anybody can fully understand (at least not in a lifetime).

There are some things that obviously can be discovered (gold, coal, species of flora/fauna) but these have existed for as long as they have, and therefore their discovery only relates to our human realisation of their existence or the science behind their existence.

So it really bools down to how you define the words you use to describe maths, because its rules still apply to the universe regardless of whether you know about them or not.
Coel Hellier - on 16 Jan 2013
In reply to Jimbo W:

> So in that case, is mathematics provisional in the same way that scientific theories about
> reality are provisional?

Yes.

> If, rather, mathematics is provisional, in what way is it provisional ...

For example, all humans (all mathematicians) could have a delusional blind-spot about some aspect of logic, and so always get something wrong. (The fact that I answered "yes" above doesn't mean that things can't be proved well beyond reasonable doubt, just as much of science, though formally provisional, is well beyond reasonable doubt.)

> So reality and maths are both agency independent?

I would say so (though noting the formally provisional nature of any such conclusion).

> Is it? Or is it a tool that we use to access it?

Both. Yes it is a tool, a tool fashioned out of empirical enquiry, in the same way that a telescope is both derived from empirical enquiry and a tool towards further enquiry.

> Is it more fundamental than that?

Is maths more fundamental than reality? I'd say no, and I'm not even sure how it could be. To me maths is a description of reality, and the description/nature of something has the same fundamentalness (if that's a word) as the thing itself.

> Why the machine metaphor ...

The brain is a tool, a "machine" is a tool.

> What sort of environmental evolutionary pressure would drive this?

Our higher mathematical ability is likely a spandrel, something not directly selected for, but a by-product of things that are selected for.
1poundSOCKS - on 16 Jan 2013
In reply to Coel Hellier: Is the brain a tool? I always think of a tool as something we use, but saying we 'use' the brain just sounds a bit wrong, I think of our consciousness as being contained in the brain, so the brain contains us, in that sense. A bit of a babble, maybe it makes some sense? :S
ads.ukclimbing.com
MG - on 16 Jan 2013
In reply to Coel Hellier:
> (In reply to Jimbo W)
>
> [...]
>
> Yes. [regarding maths as provisional]

Seems an odd way of looking at it. Often pure maths is developed that has no application but is still correct (for given axioms). How is this provisional?
Jimbo W on 16 Jan 2013
In reply to Coel Hellier:

> Yes.
> For example, all humans (all mathematicians) could have a delusional blind-spot about some aspect of logic, and so always get something wrong. (The fact that I answered "yes" above doesn't mean that things can't be proved well beyond reasonable doubt, just as much of science, though formally provisional, is well beyond reasonable doubt.)

What is the pattern of matter that inspires the right angle triangle and the subsequent description by Pythagoras? In what way can such a mathematical description be said to be provisional? Or do you mean only that mathematics, holistically, as a system and tool can be regarded provisionally?

> I would say so (though noting the formally provisional nature of any such conclusion).

Given that this is not falsifiable, on what evidence does it rest?
999thAndy on 16 Jan 2013
In reply to shaymarriott:

Newton quantified gravity, he created a good model to describe it, he didn't 'invent' it.

He did a lot of work on differential calculus (in parallel to Leibniz) so he 'discovered' some maths there. The maths (calculus in this case) stands alone and abstract from the physical world, it's just that it is the best thing we've got to model the physical world with.
Coel Hellier - on 16 Jan 2013
In reply to 1poundSOCKS:

> I always think of a tool as something we use, but saying we 'use' the brain just sounds a bit
> wrong, I think of our consciousness as being contained in the brain, so the brain contains us, in
> that sense. A bit of a babble, maybe it makes some sense?

You seem to be identifying "me" or "us" with our consciousnesses, namely only one part of the whole system or body that is "us". From the point of view of our body, the brain is one part, a tool with its own functions, and consciousness is one aspect of that tool.
Coel Hellier - on 16 Jan 2013
In reply to MG:

> Often pure maths is developed that has no application but is still correct (for given axioms).
> How is this provisional?

For starters, it depends on us having reasoned correctly from the axioms. Do you have any independent verification of that?
MG - on 16 Jan 2013
In reply to Coel Hellier:
> (In reply to MG)
>
> [...]
>
> For starters, it depends on us having reasoned correctly from the axioms. Do you have any independent verification of that?

Ah, OK. I thought you linked maths explicitly to the things that exist (sorry!), and were saying it was provisional in that it only described our understanding of things, which is itself provisional.


Coel Hellier - on 16 Jan 2013
In reply to Jimbo W:

> What is the pattern of matter that inspires the right angle triangle and the subsequent description by Pythagoras?

Those were both likely inspired by the construction of buildings in ancient Egypt, Greece or wherever.

> In what way can such a mathematical description be said to be provisional?

(1) We could have reasoned incorrectly from the axioms (no human is infallible). (2) We could have chosen axioms that are not valid in our reality.

> Given that this is not falsifiable, on what evidence does it rest?

We discussed this one extensively last time. The evidence is simply that "external universe" models are far better at explaining and prediction than "disembodied mind floating in a vacuum" models.
Coel Hellier - on 16 Jan 2013
In reply to MG:

> I thought you linked maths explicitly to the things that exist (sorry!), and were saying it was
> provisional in that it only described our understanding of things, which is itself provisional.

Well I was saying something along those lines, if you want our maths to map to our reality. Of course you could conceptualise a self-contained logical system about which you don't care whether it maps to our reality. But that is not the maths we have, which very definitely is derived from our universe. Essentially our maths is a distillate of empirical observation of our reality.

1poundSOCKS - on 16 Jan 2013
In reply to Coel Hellier: This feels like a very self-indulgent side issue, sorry OP :) Anyway, I can just see for example my hand as being a tool, more than the brain, because my brain controls it. We need a brain/consciousness thread (not that I'm volunteering to start it!).
Coel Hellier - on 16 Jan 2013
In reply to 1poundSOCKS:

> Anyway, I can just see for example my hand as being a tool, more than the brain, because my brain controls it.

And both hand and brain are tools, tools of the genes, programmed by the genes to do a job, namely to propagate genes to the next generation.
MG - on 16 Jan 2013
In reply to Coel Hellier:
> (In reply to MG)
>
> [...]
>
> Well I was saying something along those lines, if you want our maths to map to our reality. Of course you could conceptualise a self-contained logical system about which you don't care whether it maps to our reality. But that is not the maths we have,

I don't think that's right. There are mathematical descriptions of all sorts of things that don't have physical counterparts. An inverse cube law could be used to predict gravitational effects. It would nonsense physically but make mathematical sense. More generally there are many examples of pure maths being developed that don't related to reality.

1poundSOCKS - on 16 Jan 2013
In reply to Coel Hellier: And the genes were programmed by God, case solved! :)
Coel Hellier - on 16 Jan 2013
In reply to MG:

> More generally there are many examples of pure maths being developed that don't related to reality.

This is what I'd deny -- in any such pure-maths situations the vast majority of the axioms being used will have ultimately been derived from empirical reality. Thus these pure-maths conceptions are still products of empiricism.

It could indeed be the case that mathematicians would explore by holding most of the axioms as they are and exploring what happens if a subset were changed, but this is the same process as a physicist developing possible models, again based on empiricism but exploring variations.
Robert Durran - on 16 Jan 2013
In reply to MG:
> (In reply to Coel Hellier)
>
> I don't think that's right. There are mathematical descriptions of all sorts of things that don't have physical counterparts. An inverse cube law could be used to predict gravitational effects. It would nonsense physically but make mathematical sense. More generally there are many examples of pure maths being developed that don't related to reality.

I'm with you, not Coel, on this. I think.....

MG - on 16 Jan 2013
In reply to Coel Hellier: What about non-Euclidean geometries? They (and their axioms) were developed well in advance of any physical understanding of them.
cb294 - on 16 Jan 2013
In reply to Coel Hellier:

But couldnt you devise a closed axiomatic system that would let you derive theorems based on consistency with the starting set of axioms but without reference to external reality?

Possibly pointless except as a toy to study the meta level of mathematical logic.


CB
MG - on 16 Jan 2013
In reply to Robert Durran:
> (In reply to MG)
> [...]
>
> I'm with you, not Coel, on this. I think.....

I am too. I think....
Robert Durran - on 16 Jan 2013
In reply to cb294:
> (In reply to Coel Hellier)
>
> But couldnt you devise a closed axiomatic system that would let you derive theorems based on consistency with the starting set of axioms but without reference to external reality?

Aren't pure mathematicians doing this all the time? Sometimes the axioms will be inspired by reality. Sometimes they will not be and may (as in non-euclidean geometry) or may not later apply to a currently unsuspected aspect of reality.
Coel Hellier - on 16 Jan 2013
In reply to MG:

> What about non-Euclidean geometries? They (and their axioms) were developed well in
> advance of any physical understanding of them.

Yes, but they were developed by a process of (1) taking most axioms of maths and logic, (2) relaxing a few, and (3) seeing what resulted. Because of that 1 they are still products of empirical observation. Such mathematicians are acting just like theoretical physicists in exploring possible variations (and then one can consider whether the variations match reality).
Coel Hellier - on 16 Jan 2013
In reply to cb294:

> But couldnt you devise a closed axiomatic system that would let you derive theorems based on
> consistency with the starting set of axioms but without reference to external reality?

Maybe you could, starting from a blank sheet of paper and not using any axioms of (previous) maths or logic whatsoever. However I doubt whether anyone has ever done that.
Jimbo W on 16 Jan 2013
In reply to Coel Hellier:

> This is what I'd deny -- in any such pure-maths situations the vast majority of the axioms being used will have ultimately been derived from empirical reality. Thus these pure-maths conceptions are still products of empiricism.

Presumably you mean producs if sense data rather than the ism of emporicism? But that is like saying because we can't become organisms capable of abstract thought without lots of sense data, any of our articulations about reality are necessarily empirical in nature. This is very reminiscent to your way of defining existence with respect to potential chains of causal connections to our sense data. When does any human articulation start and stop being "empirical"? I can't see the limits and therefore the meaning in the way that you are using the word with regard to mathematics.

MG - on 16 Jan 2013
In reply to Coel Hellier:
> (In reply to MG)
>
> [...]
>
> Yes, but they were developed by a process of (1) taking most axioms of maths and logic, (2) relaxing a few, and (3) seeing what resulted. Because of that 1 they are still products of empirical observation.

Well I would say that because of 2) they are not! They be influenced by empirical observations but are no longer products of them. Also the key difference with theoretical physics is that mathematicians (certain with these geometries) had no real interest in seeing which matched reality. That came much later.
Coel Hellier - on 16 Jan 2013
In reply to Jimbo W:

> But that is like saying because we can't become organisms capable of abstract thought without lots
> of sense data, any of our articulations about reality are necessarily empirical in nature.

Yep!

> This is very reminiscent to your way of defining existence with respect to potential chains of causal connections to our sense data.

Yep!!

> When does any human articulation start and stop being "empirical"?

It doesn't, it's empiricism all the way down.
Coel Hellier - on 16 Jan 2013
In reply to MG:

> Well I would say that because of 2) they are not! They be influenced by empirical observations
> but are no longer products of them.

The non-Euclidean geometries come from asking "is this particular axiom necessary?", while keeping everything else. It's exactly the sort of thinking about variations of current models that physicists do.

> Also the key difference with theoretical physics is that mathematicians (certain with these geometries)
> had no real interest in seeing which matched reality. That came much later.

I don't see that that makes much difference, they were still exploring patterns, where their exploration was ultimately grounded in empirical reality.
MG - on 16 Jan 2013
In reply to Coel Hellier:
> (In reply to MG)
>
> [...]
>
> The non-Euclidean geometries come from asking "is this particular axiom necessary?",

Quite. But not, "is this particular axiom necessary and do the results still accord with reality".
>
> I don't see that that makes much difference, they were still exploring patterns,

Yep.

where their exploration was ultimately grounded in empirical reality.

Nope. Well it turns it was in many cases but mathematicians of the time didn't know that. So you could argue the other way around - that physicists spend their time trying to fit reality to maths.
Jimbo W on 16 Jan 2013
In reply to Coel Hellier:

> It doesn't, it's empiricism all the way down.

But this is not discriminatory, in what way is apparently useless maths different from say Sheldrake's theory of morphic resonance? Both, are based on "empiricism", given that its "empiricism all the way down". By having it such, all discriminatory and meaningful power seems to me to be lost.
Jimbo W on 16 Jan 2013
In reply to Coel Hellier:

> It doesn't, it's empiricism all the way down.

And is not what a bacterium does in "sensing" and subsequently "responding" to its environment also "empirical" in this broad sense?
Dave Garnett - on 16 Jan 2013
In reply to Robert Durran:
> (In reply to cb294)
> [...]
>
> Aren't pure mathematicians doing this all the time? Sometimes the axioms will be inspired by reality. Sometimes they will not be and may (as in non-euclidean geometry) or may not later apply to a currently unsuspected aspect of reality.

I seem to recall that Einstein (and Dirac, possibly) dismissed mathematical solutions that didn't make sense (like negative values from taking square roots) only for it to turn out that these solutions did, indeed, have real world (or real universe anyway) meaning.

Jimbo W on 16 Jan 2013
In reply to MG:

> Well I would say that because of 2) they are not! They be influenced by empirical observations but are no longer products of them. Also the key difference with theoretical physics is that mathematicians (certain with these geometries) had no real interest in seeing which matched reality. That came much later.

I agree. Furthermore, with such a liberal attitude to what consitutes an empirical basis, absolutely everything that a human or animal produces can be regarded as empirical. The fact that mathematical ideas often precede any application, if there is one, to physical reality, also suggests that such ideas must also be at least to some extent invented as they appear as ideas in the brain.
Coel Hellier - on 16 Jan 2013
In reply to Jimbo W:

> The fact that mathematical ideas often precede any application, if there is one, to physical reality, also
> suggests that such ideas must also be at least to some extent invented as they appear as ideas in the brain.

But that's also how physics and the rest of science works. Scientists dream up ideas, then test them. It's still the case that scientific claims are empirical. Whether the gap to testing is 10 mins or 200 years, and whether it is done by the same person or not, doesn't really change things.
Coel Hellier - on 16 Jan 2013
In reply to Jimbo W:

> And is not what a bacterium does in "sensing" and subsequently "responding" to its environment
> also "empirical" in this broad sense?

Yes!

Coel Hellier - on 16 Jan 2013
In reply to Jimbo W:

> But this is not discriminatory, in what way is apparently useless maths different from say
> Sheldrake's theory of morphic resonance?

The difference is seen by considering the claims that are being made. Sheldrake make claims about reality that can be empirically tested.

In the case of maths you have two types of claims: applied maths, which is also making claims about reality that can be empirically tested, or pure maths. The pure-maths claims are of the form "if {axioms} then {result}". The only claim being made is about logic, but the only validation of that logic is empirical.
Jimbo W on 16 Jan 2013
In reply to Coel Hellier:

> But that's also how physics and the rest of science works. Scientists dream up ideas, then test them. It's still the case that scientific claims are empirical. Whether the gap to testing is 10 mins or 200 years, and whether it is done by the same person or not, doesn't really change things.

Well I disagree. I think if the mathematics is not explicitly based in reality, it must be invented. Afterall, what is the idea about? If the idea is about some aspect of reality, then all available information will inevitably be brought to bare on the thinking about the aspect and the ideas that emerge, but if the mathematics is a pure indulgence for the sake say indulgent aesthetics, then while it is the product of an "empirical" beast (in the broad sense of your use of the word) I think this position is distinguishable from the way that theories emerge about science in that such maths does not have an immediate empirical coordinate, and no necessary application toward reality.
Jim Fraser - on 16 Jan 2013
In reply to Jimbo W:

Discovered.

Number or quantity and the relationships between them exist without our agency and are there to be discovered.


(As an ex-Napier guy I can find myself referring to John Napier as the man who discovered logarithms and I must say that I enjoy the strange looks and discussion that result.)
Coel Hellier - on 16 Jan 2013
In reply to Jimbo W:

> Well I disagree. I think if the mathematics is not explicitly based in reality, it must be invented.

Perhaps we need to clarify what we're disagreeing about. In arguing that maths is "empirical" I'm not denying that a particular theory can be "invented". For comparison, the steam engine was invented, but the whole invention was utterly steeped in empirical enquiry and could not have been done except from a thoroughly empirical grounding.

> ... I think this position is distinguishable from the way that theories emerge about science in that
> such maths does not have an immediate empirical coordinate, and no necessary application toward reality.

Perhaps you're underestimating the extent to which theoretical physicists can pursue "a pure indulgence for the sake say indulgent aesthetics" without caring about any "immediate empirical coordinate". ;-) String theory is a good example.
Jimbo W on 16 Jan 2013
In reply to Coel Hellier:

> Perhaps we need to clarify what we're disagreeing about. In arguing that maths is "empirical" I'm not denying that a particular theory can be "invented". For comparison, the steam engine was invented, but the whole invention was utterly steeped in empirical enquiry and could not have been done except from a thoroughly empirical grounding.

Okay. What about Godel's incompleteness theorems? In what way are they:
a) empirical?
b) discovered vs invented?
cb294 - on 16 Jan 2013
In reply to Jim Fraser:
> (In reply to Jimbo W)
>
> Discovered.
>
> Number or quantity and the relationships between them exist without our agency and are there to be discovered.
>
>

But the particular notation can be invented.

CB

Denni on 16 Jan 2013
In reply to Jimbo W:

I've just discovered how to do multiplication the chinese way, very interesting!
As for the OP's question, not knowledgeable enough to comment.

Good luck to you all :0)
Jimbo W on 16 Jan 2013
In reply to Coel Hellier:

> Perhaps you're underestimating the extent to which theoretical physicists can pursue "a pure indulgence for the sake say indulgent aesthetics" without caring about any "immediate empirical coordinate". ;-) String theory is a good example.

> > ... I think this position is distinguishable from the way that theories emerge about science in that such maths does not have an immediate empirical coordinate, and no necessary application toward reality.

> Perhaps you're underestimating the extent to which theoretical physicists can pursue "a pure indulgence for the sake say indulgent aesthetics" without caring about any "immediate empirical coordinate". ;-) String theory is a good example.

Quite the opposite! I have a healthy disdain for such theories precisely because it does seem to have lost its "immediate empirical coordinate" :)
Indeed, useful as quantum theories are, I'd go far further and side with both Einstein and Penrose in wanting to take a realist position on what science can say and how it should be expressed, rather than allow the idealism of theoretical sufficiency to suffice.
Robert Durran - on 16 Jan 2013
In reply to Coel Hellier:
> (In reply to Jimbo W)

> Perhaps you're underestimating the extent to which theoretical physicists can pursue "a pure indulgence for the sake say indulgent aesthetics" without caring about any "immediate empirical coordinate".

But don't they pursue the aesthetic theories because they believe (from previous experience) that successful theories tend to be the mathematically elegant ones.

Jimbo W on 16 Jan 2013
In reply to Robert Durran:

> But don't they pursue the aesthetic theories because they believe (from previous experience) that successful theories tend to be the mathematically elegant ones.

I think there's a lot of truth in that. I remember a debate in which Freeman Dyson criticised Hawking because he's forgotten that there's a difference between theory and reality.
Robert Durran - on 16 Jan 2013
In reply to Jimbo W:
> (In reply to Coel Hellier)
> Indeed, useful as quantum theories are, I'd go far further and side with both Einstein and Penrose in wanting to take a realist position on what science can say and how it should be expressed, rather than allow the idealism of theoretical sufficiency to suffice.

But didn't Einstein also have great confidence in General Relativity being correct because of its elegance.

lithos on 16 Jan 2013

'Mathematics is the subject in which you don't know what you are
talking about and don't care whether what you say is true'

Coel Hellier - on 16 Jan 2013
In reply to Jimbo W:

> Okay. What about Godel's incompleteness theorems? In what way are they:
> a) empirical?

The axioms that he reasoned from are empirically derived.

> b) discovered vs invented?

I'd say that the incompleteness consequence of those axioms was "discovered" rather than invented. Though, as above, I'm not sure there has ever been any "invention" that was not thoroughly grounded in empiricism, so I'm not sure that the discovered/invented distinction amounts to much.
Jimbo W on 16 Jan 2013
In reply to Robert Durran:

> But didn't Einstein also have great confidence in General Relativity being correct because of its elegance.


From what I remember of what I've read of Einstein, he definitely did have great confidence in the correctness of General Relativity, not because of mathematical elegance, but rather because of explanatory power and what he perceived as an apparent revelation of reality. And didn't Einstein say:

If you are out to describe the truth, leave elegance to the tailor.

With regard to this discussion, its pertinent to provide a link to Einstein's discussion of geometry:
http://www-history.mcs.st-and.ac.uk/Extras/Einstein_geometry.html
Jimbo W on 16 Jan 2013
In reply to Coel Hellier:

> > Okay. What about Godel's incompleteness theorems? In what way are they:
> > a) empirical?

> The axioms that he reasoned from are empirically derived.

Which axioms? The Peano axioms? Can you show how they are all empirically grounded?
http://www.research.ibm.com/people/h/hirzel/papers/canon00-goedel.pdf

> > b) discovered vs invented?

> I'd say that the incompleteness consequence of those axioms was "discovered" rather than invented. Though, as above, I'm not sure there has ever been any "invention" that was not thoroughly grounded in empiricism, so I'm not sure that the discovered/invented distinction amounts to much.

If they are discovered, then does that mean they are written into the fabric of reality? I noticed before that you didn't directly answer my question on Tegmark above, but wonder in what sense mathematics is a facet of reality without our necessary presence to discover it?
Coel Hellier - on 16 Jan 2013
In reply to Jimbo W:

> Which axioms? The Peano axioms? Can you show how they are all empirically grounded?

Where else do we get Peano's axioms from other than the fact that they work? They are not a merely arbitrary set plucked out of nothing.

> in what sense mathematics is a facet of reality without our necessary presence to discover it?

See my first post in this thread for my opinion on that.
mgco3 - on 16 Jan 2013
In reply to Jimbo W: Mathematics has been around for a long time and we are still learning.

The mayan calendar debacle has taught us on thing, if your calendar isn't right, its not the end of the world..
Jimbo W on 16 Jan 2013
In reply to Coel Hellier:

> Where else do we get Peano's axioms from other than the fact that they work? They are not a merely arbitrary set plucked out of nothing.

So, firstly, let me get this straight.. ..your view of "empirically grounded" is equal to "the fact that they work"? Do you agree and could you expand? While I would agree that natural numbers appear to work intuitively, I'm not so clear as to how the Peano axioms are empirically based, not least because of, for example, Russell's criticisms Peano's undefined terms: 0, number and successor, and the ease of mutation of the axiomatic products, e.g. see here:

http://books.google.co.uk/books?id=SoqEipeoO_cC&pg=PT171&lpg=PT171#v=onepage&q&f=fal...

So they don't necessarily "work".
Jimbo W on 16 Jan 2013
In reply to Coel Hellier:

> See my first post in this thread for my opinion on that.

The literal existence of mathematics appears to me an absurd concept in the absence of agency unless one takes an radical Platonist position vis a vis Tegmark's mathematical universe, or alternatively a pantheistic type views in which maths is the language expressing the mind of god and one can express a "pantheistic" faith about its existence. In any other sense, such a position appear to me to be totally barmy.
Coel Hellier - on 16 Jan 2013
In reply to Jimbo W:

> So, firstly, let me get this straight.. ..your view of "empirically grounded" is equal to "the fact that they work"?

Yes.

> I'm not so clear as to how the Peano axioms are empirically based, not least because of, for example,
> Russell's criticisms Peano's undefined terms:

OK, I'll rephrase slightly. Mathematics works, indeed it works very well (and all of engineering and technology is based on it, proving that it works), and thus the axioms of maths work.

(By that rephrasing I avoid discussion of Peano's formulation in particular, which I'm not expert on.)
Coel Hellier - on 16 Jan 2013
In reply to Jimbo W:

> The literal existence of mathematics appears to me an absurd concept in the absence of agency ...

The existence of patterns is not absurd in the absence of agency, though the study of those patterns does of course require agency.

> vis a vis Tegmark's mathematical universe,

I'm guessing that by "everything is maths" Tegmark meant "everything is patterns" rather than "everything is the study of patterns".
Jimbo W on 17 Jan 2013
In reply to Coel Hellier:
> (In reply to Jimbo W)
>
> [...]
>
> Yes.
>
> [...]
>
> OK, I'll rephrase slightly. Mathematics works, indeed it works very well (and all of engineering and technology is based on it, proving that it works), and thus the axioms of maths work. (By that rephrasing I avoid discussion of Peano's formulation in particular, which I'm not expert on.)

That's a big rephrase. My question was about the empirical basis for Godel's incompleteness theorems and you have now dissolved back to the generalism of "mathematics works". So you concede that aspects of mathematics do not directly have empirical coordinates, but you reckon them nevertheless empirical by virtue of some distal property of physical reality that permits the abstraction to numbers.
Jimbo W on 17 Jan 2013
In reply to Coel Hellier:

> I'm guessing that by "everything is maths" Tegmark meant "everything is patterns" rather than "everything is the study of patterns".

No. Tegmark's hypothesis denies the physical reality of the universe making it rather subordinate to mathematics. The universe *is* mathematics at root. And where complex enough mathematical substructures exist that are self aware, they will perceive themselves as having a *physical* reality - the latter being as I understand it illusory.
Coel Hellier - on 17 Jan 2013
In reply to Jimbo W:

> That's a big rephrase. My question was about the empirical basis for Godel's incompleteness
> theorems and you have now dissolved back to the generalism of "mathematics works".

I'd have to look up which axioms Godel reasoned from, but my assertion is that those axioms would not have been an arbitrary ad-hoc selection chosen for no reason, but were adopted because they work in our universe. In that sense, Godel's theorems result from empiricism.
Jimbo W on 17 Jan 2013
In reply to Coel Hellier:
> (In reply to Jimbo W)
>
> [...]
>
> The existence of patterns is not absurd in the absence of agency, though the study of those patterns does of course require agency.

Yes, but you are rendering patterns synonymous with mathematics. Does maths = patterns, or does maths = the description / abstraction of those patterns. If maths = patterns, on what way is that so, and in what way is our concept of mathematics relevant to the behaviour in physical reality? Matter does what it does... ...what need has it if of maths? If maths is somehow embodied in the behaviour of matter, then what is the basis if this rule following behaviour? What in matter invokes such order?
Jimbo W on 17 Jan 2013
In reply to Coel Hellier:

> I'd have to look up which axioms Godel reasoned from, but my assertion is that those axioms would not have been an arbitrary ad-hoc selection chosen for no reason, but were adopted because they work in our universe. In that sense, Godel's theorems result from empiricism.

I gave you a link to the translation of his original paper above. The axioms are not ad hoc, the peano axioms are devised to provide the establishment of the natural numbers. As I said, it is not the concordance intended, but the empirical basis on which the axioms and definitions rest that is under question.
Robert Durran - on 17 Jan 2013
In reply to Jimbo W:
> (In reply to Robert Durran)
> From what I remember of what I've read of Einstein, he definitely did have great confidence in the correctness of General Relativity, not because of mathematical elegance, but rather because of explanatory power and what he perceived as an apparent revelation of reality.

I may be wrong, but I thought Einstein came up with GR as an extension of SR through pure thought and hunches/insights such as the indistinguishability of acceleration and a gravitational field, and was convinced by its consistency and elegance rather than its ability to explain any previous observations incompatible with Newtonian Gravity. It did, of course, allow correct predictions to be made and confirmed by observation later. Coel will know!
Robert Durran - on 17 Jan 2013
In reply to Jimbo W:
> (In reply to Coel Hellier)
The axioms are not ad hoc, the peano axioms are devised to provide the establishment of the natural numbers.

So did natural numbers "exist" before Peano came up with his axioms?!
Jimbo W on 17 Jan 2013
In reply to Robert Durran:
> (In reply to Jimbo W)
> [...]
> The axioms are not ad hoc, the peano axioms are devised to provide the establishment of the natural numbers.
>
> So did natural numbers "exist" before Peano came up with his axioms?!

I'd retort with the question, do natural numbers exist in the absence of agency to describe / define them. If I answer yes to this question than I would have to go the whole hog and accept tegmark's hypothesis. If I answer no, then maths is an invention which we find useful to describe the behaviour of matter.
Coel Hellier - on 17 Jan 2013
In reply to Jimbo W:

> Yes, but you are rendering patterns synonymous with mathematics.

Yes.

> Does maths = patterns, or does maths = the description / abstraction of those patterns.

As meant in Tegmark's phrase, etc, maths = patterns (rather than the study of those patterns).

> Matter does what it does... ...what need has it if of maths?

The "what matter does" *is* the pattern and thus is the maths. (You're right that matter doesn't need maths in the "study of ..." sense.)

> If maths is somehow embodied in the behaviour of matter, then what is the basis if this rule
> following behaviour? What in matter invokes such order?

As far as we can tell, matter has properties; those properties are about how it interacts with other matter, and thus determine what patterns it makes.

Why is matter ordered? Why does it have properties that can be described by simple rules? Hmm, well those are rather profound questions and I don't know the answer to it. What's the alternative? A universe in which matter showed no order or regularity, and in which there was thus no pattern, only chaos. I guess another alternative is nothing at all.
Coel Hellier - on 17 Jan 2013
In reply to Jimbo W:

> The axioms are not ad hoc, the peano axioms are devised to provide the establishment of the natural numbers.

OK, and surely the natural numbers are an empirical construct, derived from the nature of our universe. Thus, if the Peano's axioms are essentially a distillate of natural numbers, then they are a distillate of empirical enquiry.
Coel Hellier - on 17 Jan 2013
In reply to Jimbo W:

> I'd retort with the question, do natural numbers exist in the absence of agency to describe / define them.

Steering somewhat clear of getting into what "exist" means (!), I'd say that the behaviour and regularities for which natural numbers are a description certainly do exist in the absence of agency to describe them.

In that sense the "natural numbers" are not arbitrary, but are something that anyone describing how the universe is would arrive at. So, if by "natural numbers" we mean the patterns that show natural-number properties, then yes they do exist independently of us.
Robert Durran - on 17 Jan 2013
In reply to Jimbo W:
> (In reply to Robert Durran)
> I'd retort with the question, do natural numbers exist in the absence of agency to describe / define them. If I answer yes to this question than I would have to go the whole hog and accept tegmark's hypothesis.

I'd like to answer yes, but going the whole hog......?

> If I answer no, then maths is an invention which we find useful to describe the behaviour of matter.

So matter just does its "own thing" and maths/physics is just our bumbling attempt to describe the best we can what it's doing....

Jimbo W on 17 Jan 2013
In reply to Robert Durran:

> I'd like to answer yes, but going the whole hog......?

Well, if maths is derivative of matter's behaviour, then it begs the question what on earth makes matter follow such order. The alternative seems to me to say that maths is the reality, and physical expressions of it that we perceive are illusions that give our self awareness meaning.

> So matter just does its "own thing" and maths/physics is just our bumbling attempt to describe the best we can what it's doing....

Yes, pretty much. I mean, look at the language we use, e.g. "Law" a thoroughly anthropocentric, if not theocentric construct!
Jimbo W on 17 Jan 2013
In reply to Robert Durran:

Indeed, in the application of words like "law", it isn't even a construct so much as a metaphor.
Jimbo W on 17 Jan 2013
In reply to Robert Durran:

> So matter just does its "own thing" and maths/physics is just our bumbling attempt to describe the best we can what it's doing....

Furthermore, this hints at one of the gross illusions of science... ...that it explains anything. It describes the behaviour very well, it formalises those things into equations, but does it really explain anything? See Coel's recent blog on "why" questions and the idea that science addresses "how" questions, while religion addresses "why" questions. This post judiciously avoids the real nub of the problem. To ask a why question isn't to necessarily invoke the requirement for ultimate agency and purposive answers, but it is a question posed because of the recognition of the problematic nature of an aspect of physical subject under scrutiny. Why is there something rather than nothing? Why us physical reality ordered? Why does matter either create or follow order? Why is there an order to be followed? Why do we think that order has a real ontological status etc?
Robert Durran - on 17 Jan 2013
In reply to Jimbo W:
> (In reply to Robert Durran)
>
> Furthermore, this hints at one of the gross illusions of science... ...that it explains anything. It describes the behaviour very well, it formalises those things into equations, but does it really explain anything?

By going with Tegmark, it at least offers the hope that the equations are more than just a rough description of what happens (which would be nice!), but it does, of course, beg the question of where the maths comes from. It would be hard to imagine ever knowing.

Jimbo W on 17 Jan 2013
In reply to Robert Durran:

> By going with Tegmark, it at least offers the hope that the equations are more than just a rough description of what happens (which would be nice!), but it does, of course, beg the question of where the maths comes from. It would be hard to imagine ever knowing.

Well, I don't know, they only reason for feeling that is because we regard maths as abstractive rather than substantive and basic. If it were the latter, then would the question "where does it come from?" actually have any meaning? I mean it seems to me qualitatively different from the conventional materialistic scientific worldview in which Hawking's "because there is a law such as gravity, the universe can and will create itself from nothing", is non sensical involving both the presence of "law" and that which the law describes, begging the question of where these things come from. People are often tempted to attack the proposal of god as a creator with the "well, what caused god" approach. The answer being nothing, because the question literally doesn't make any sense, because causation is itself created. Similarly, if our reality is a subordinate illusion within a mathematical reality, causation is also an illusion that itself needs no explanation, and thus neither does maths.
Coel Hellier - on 17 Jan 2013
In reply to Jimbo W:

> Why is there something rather than nothing? Why us physical reality ordered?

Interesting questions, though I don't have answers.

> Why does matter either create or follow order? Why is there an order to be followed?

I don't think we should separate matter from order in thinking about this. The "order" is simply the nature of matter (it isn't a separate thing that is either imposed on or derived from matter).
Robert Durran - on 17 Jan 2013
In reply to Jimbo W:
> (In reply to Robert Durran)
> Well, I don't know, the only reason for feeling that is because we regard maths as abstractive rather than substantive and basic. If it were the latter, then would the question "where does it come from?" actually have any meaning?

And maths IS reality? But if the former, does it in fact lie outside reality (with matter and/or physical law being reality)

Jimbo W on 17 Jan 2013
In reply to Robert Durran:

> And maths IS reality? But if the former, does it in fact lie outside reality (with matter and/or physical law being reality)

Well, no, I don't think it can be "outside", because we can clearly appreciate, utilise and describe it, so we can at least relate to it within what we think is our reality, unless, of course, you are in some way making mathematics some kind of characteristic of a transcendent god.
Jimbo W on 17 Jan 2013
In reply to Coel Hellier:

> Interesting questions, though I don't have answers.

But to at least to attempt to answer these questions would be to attempt to explain, rather than describe and codify the mysteries of our reality. Are you saying you can't have any answers? If not, why not? Do you think it's turtles all the way down?

> I don't think we should separate matter from order in thinking about this. The "order" is simply the nature of matter (it isn't a separate thing that is either imposed on or derived from matter).

Well, I think that is an interesting theory, but I don't think that that is necessarily true. Is order an intrinsic character of matter (what do we mean by matter here) or is matter an intrinsic characteristic of order?
Robert Durran - on 17 Jan 2013
In reply to Jimbo W:
> (In reply to Coel Hellier)
> Is order an intrinsic character of matter (what do we mean by matter here)...... or is matter an intrinsic characteristic of order?

Is that the same as as saying matter inevitably emerges as an illusion once consciousness has emerged from the fundfamental order/patterns/maths?

Jimbo W on 17 Jan 2013
In reply to Robert Durran:

> Is that the same as as saying matter inevitably emerges as an illusion once consciousness has emerged from the fundfamental order/patterns/maths?

Not necessarily, but it is one possibility.
Coel Hellier - on 17 Jan 2013
In reply to Jimbo W:

> Are you saying you can't have any answers?

Maybe we can answer those questions, maybe we can't. I don't know the answers to those ones, and I don't know whether we can have an answer.

> Is order an intrinsic character of matter (what do we mean by matter here)

I'd answer yes to that (at least order is an intrinsic part of the matter in our universe). The only way of specifying what matter "is" is an account of how it interacts with other things, and that account is an account of the patterns/order that it leads to.

> or is matter an intrinsic characteristic of order?

That would seem a weird way of phrasing it to me, since it implies that "order" has some "existence" in its own right, whereas I think of it as a property of matter.
Robert Durran - on 17 Jan 2013
In reply to Coel Hellier:
> (In reply to Jimbo W)
> That would seem a weird way of phrasing it to me, since it implies that "order" has some "existence" in its own right, whereas I think of it as a property of matter.

Which is another way of saying that order/pattern/maths is a property of matter to be discovered not invented?

Coel Hellier - on 17 Jan 2013
In reply to Robert Durran:

> Which is another way of saying that order/pattern/maths is a property of matter to be discovered not invented?

Yes.

But as I said above I'm not sure that the discovered/invented distinction amounts to much. You can't invent things arbitrarily, you can only invent things that go with the grain of the universe, not against, so you can't invent perpetual motion machines or wings that you can strap on your back and allow you to flap your arms and fly.
Robert Durran - on 17 Jan 2013
In reply to Coel Hellier:
> (In reply to Robert Durran)
> You can only invent things that go with the grain of the universe, not against.

Maybe I could invent an i x pi dimensional cube. Would that go with the grain of the universe?
Coel Hellier - on 17 Jan 2013
In reply to Robert Durran:

> Maybe I could invent an i x pi dimensional cube. Would that go with the grain of the universe?

Hmm, I'm not sure. Could you knock one up out of meccano and see what it looks like?
Jimbo W on 17 Jan 2013
In reply to Coel Hellier:

> That would seem a weird way of phrasing it to me, since it implies that "order" has some "existence" in its own right, whereas I think of it as a property of matter.

Why shouldn't it exist in its own right? It may not be intuitive, but neither were Maxwell's fields, Einstein's relativities and Quantum Mechanics not so very long ago. Why should order be subordinate to matter, and not vice versa? What is the evidence that it is subordinate to matter?
Robert Durran - on 17 Jan 2013
In reply to Coel Hellier:
> (In reply to Robert Durran)
>
> Hmm, I'm not sure. Could you knock one up out of meccano and see what it looks like?

No, but it is easy to describe a, say, 50 dimensional one mathematically. That probably goes against the grain of the universe too - and if the universe turns out to be 50 dimensional, it's trivial to add a few more dimensions!

Coel Hellier - on 17 Jan 2013
In reply to Jimbo W:

> Why shouldn't it [order] exist in its own right?

Because "order" is an adjective, it describes something. Suggesting that it can exist in its own right is like suggesting that "green" or "tall" or beautiful" can exist in its own right, not just as an attribute of something. And that seems weird to me.
Jimbo W on 17 Jan 2013
In reply to Coel Hellier:

> Because "order" is an adjective, it describes something. Suggesting that it can exist in its own right is like suggesting that "green" or "tall" or beautiful" can exist in its own right, not just as an attribute of something. And that seems weird to me.

"Ordered" is the adjective, "order" is a noun. Do you think "mathematics" is an adjective too?!
Jimbo W on 17 Jan 2013
In reply to Coel Hellier:
> (In reply to Jimbo W)
>
> [...]
>
> Because "order" is an adjective, it describes something. Suggesting that it can exist in its own right is like suggesting that "green" or "tall" or beautiful" can exist in its own right, not just as an attribute of something. And that seems weird to me.

Why shouldn't matter be subordinate to order? And what is the evidence that it is order that emerges from matter rather than vice versa?
Coel Hellier - on 17 Jan 2013
In reply to Jimbo W:

> "Ordered" is the adjective, "order" is a noun. Do you think "mathematics" is an adjective too?!

Yes, I think of both "order" and "maths" as referring to patterns of stuff, rather than as entities that exist in their own right.

> Why shouldn't matter be subordinate to order?

Because I don't understand what that would even mean.

> And what is the evidence that it is order that emerges from matter rather than vice versa?

Order, or patterns of stuff, derives from the properties of the stuff. I don't see how you can consider either alone or make either as more primary than the other. It seems to me that "properties of the stuff" (and hence patterns/order) is so indistinguishable from "the stuff" that they are the same thing.
hamish - on 17 Jan 2013
In reply to Jimbo W:

I love the fact that you can find stuff like this on a climbing forum!

I came across this a while back, its old but quite good reading :

"The Unreasonable Effectiveness of Mathematics in the Natural Sciences"

http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
Robert Durran - on 18 Jan 2013
In reply to hamish:
> (In reply to Jimbo W)
>
> "The Unreasonable Effectiveness of Mathematics in the Natural Sciences"
>
> http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

Great quote from Bertrand Russell at the top of the article:

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty, a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry."

I think Russell, however, in later life became somewhat less enthused with the beauty of maths, regarding it as just one big tautology. Shame really.


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