Question for those that consider themselves mathematicians.

Apperception

Hi there;

As a straw poll, I'd like to get an idea of how many of those that consider themselves to be mathematicians, know what the study of mathematics actually is.

The reason for my enquiry is not to find out what mathematics is, since his was settled definitively centuries ago, and for those interested, it is left as a very non-trivial exercise to discover by themselves.

The enquiry is designed with 3 purposes in mind.

The first is to discover how many mathematicians are simply doing the engineering bit (following the rules they have been taught to get to the ends they have been told are valuable) without any insight into what any of it means.

The second is to identify individuals that do have an understanding of the underlying conceptual framework and perhaps begin something of an interesting dialogue about what is commonly understood.

The third purpose of the enquiry implicit, and it is to help those from category one above, find their way into category two.

So that confusion is kept to a minimum and the question is plain, I'll formulate the same question in different but straightforward ways.

1. What exactly is mathematics the study of?

2. What exactly is geometry the study of?

3. What exactly is algebra the study of?

4. Why is mathematics in general necessarily applicable to the descriptions of all phenomena? (Do you know why it is so useful?)

For those that have studied logic or philosophy:
5. Why can a psychologistic explanation never provide the universal necissity that mathematics would appear to have established?

6. Can you explain whether mathematics is analytic or synthetic?

Go on then if you think you're hard enough

A.

Son Goku

Now this is a thread!

I'll attempt some "off-the-cuff" answers first.

1. What exactly is mathematics the study of?

This can be answered in both the pragmatic and abstract sense. Pragmatically I would see it as the study of the manipulation of predefined conceptual entities, as well as the discovery of structural similarity of these entities leading to improved handling of the entities by analogy. Occasionally, if you are exceptionally gifted, it involves the creation of said entities.

2. What exactly is geometry the study of?

Relationships and structures within "smooth" sets. Where smooth is defined by certain invariants and historical agreement. Commonly, perhaps nearly always, the most general meaning of smooth means the set is a manifold.

3. What exactly is algebra the study of?

Algebra is the study of the basic consequences and relations forced by operations on sets.

4. Why is mathematics in general necessarily applicable to the descriptions of all phenomena? (Do you know why it is so useful?)

I'm not sure I could answer that. I could attempt a description, but I'm not sure it would explain why the gauge groups in QFT are so simple.

Anyway, I don't know what level this thread is aiming for so I'll leave it there for now and modify my answers later.

Apperception

Hey.

Sorry for taking so long, life, death etc. interfers with important matters.

I guess this forum has no other takers

Son Goku wrote:

This can be answered in both the pragmatic and abstract sense. Pragmatically I would see it as the study of the manipulation of predefined conceptual entities, as well as the discovery of structural similarity of these entities leading to improved handling of the entities by analogy. Occasionally, if you are exceptionally gifted, it involves the creation of said entities.

So it's an exercise in the manipulation of abstractions then?
Just like day dreaming but about a set of predefined abstract entities like goblins, hobbits or dragons? I guess Tolkien was an exceptionally gifted mathematician.

Sorry.

Son Goku wrote:

Relationships and structures within "smooth" sets. Where smooth is defined by certain invariants and historical agreement. Commonly, perhaps nearly always, the most general meaning of smooth means the set is a manifold.

Euclid never mentioned sets. Does this mean he was not studying geometry, or somehow did not know what he was doing?
That geometric concepts can be expressed by set theory does not mean that geometry is the study of sets. Category theory might yield the same thing. What is happen here is thinking geometry 'through' set / category theory.

An analysis of the fundamental geometric concept of extension does not yield any idea of a set, but it does require a space in which to posit the degree of extension.

Agreed though, is that analysis of all spatial concepts yields a manifold, and herein lies the clue to what is being studied in geometry, namely the means by which concepts which entail an apriori* spatial manifold are constructed.

apriori* == necessary and given universally, regardless of the particulars of the concept under consderation. e.g a triangle or line cannot be thought without the spatial manifold as the underlying condition of the construction of the object.

Now what is it that constructs these concepts (spatial or otherwise), that we might be considering?

Son Goku wrote:

Algebra is the study of the basic consequences and relations forced by operations on sets.

Again algebra is older than set theory.
Specifically it is the study of how proportions of degree are related in the construction of a concept.

a = 2b etc.

The concept of the degree of 'a' "can be constructed" by relating the degree of b to successive operations of addition.

Son Goku wrote:

I'm not sure I could answer that. I could attempt a description, but I'm not sure it would explain why the gauge groups in QFT are so simple.

I have nothing to say about simplicity, however:

Since it is the mind that constructs concepts, and the spatial manifold is an apriori* requirement for the construction of spatial concepts, then all constructions which relate to the manifold itself alone, will be necessary and universally valid.

This is why mathematics is applicable to all empricial phenomena, because when they are considered (constructed as concepts) they will necessarily contain the mind-given* spatial manifold as a condition of their construction.

mind-given* does not imply that an external spatial manifold does not exist in itself (whatever it may be like), just that it (in itself) should be distinguished from what is used in the construction of spatial concepts.

Son Goku

I won't respond to your post above just now, rather I'd like to clarify something from the OP.

Apperception wrote:

The first is to discover how many mathematicians are simply doing the engineering bit (following the rules they have been taught to get to the ends they have been told are valuable) without any insight into what any of it means.

The second is to identify individuals that do have an understanding of the underlying conceptual framework and perhaps begin something of an interesting dialogue about what is commonly understood.

Do you equate an understanding of the underlying conceptual framework with an ability to express the framework in a philosophically sound way?

For instance does my inability to give a definition of geometry that doesn't use modern mathematical terminology indicate some deficiency in my understanding of it?

What kind of definitions do you want? I suspect you want something that basically amounts to explaining the underlying similarities of the mental objects considered when one is doing geometry, that is independant of the current terminological basis.

Just one question about your actual post now:

Specifically it is the study of how proportions of degree are related in the construction of a concept.

How do algebras of functions work in this definition? Or algebras of operators?

Professor_Fink

Hi Apperception,

I hope you don't mind me asking, but are you by any chance a philosophy student? Your questions seem to have a distinct 'mathematicians are ignorant of mathematics' bent.

Basically, mathematics is a structure built upon formal logic with the addition of certain specific axioms. The existance of hobbits and goblins are not one of it's axioms, and so, no, Tolkein would not be counted as a mathematician.

Son Goku is right in saying that algebra is a study of relations with in a set together with some operators. Addition is not necessarily one of these.

Also the fact that physics is described by mathematics is largely influenced that nature was around long before mathematics was developed. Mathematics hasd grown out of the attempt of humans to describe the world around them quantatively. Sure, it has gone beyond that now, but they were its roots.

Apperception

Professor_Fink wrote:

Hi Apperception,
I hope you don't mind me asking, but are you by any chance a philosophy student? Your questions seem to have a distinct 'mathematicians are ignorant of mathematics' bent.

Hi Fink,

Yes, I read philosophy, logic, mathematics and physics.
I follow mathematical processes to construct concepts.
When those concepts contain dynamical constructs, they intend to represent possible physical phenomena.
In this case of physics the intentionality is ubiquituously conscious, but there would appear to be a mental block (among some mathematicians) when it comes to the intentionality of non dynamical mathematical constructions.

However, the relation of my academic titles to this thread, is itself academic, and cannot shed any light on the subject under discussion, unless there are readers simple minded enough to fall prey to some psychologically fallacious position like an appeal to fashionable authority, or some snobby version of common sense. We'd be better off without them anyway.

Yes. I do believe that a lot of mathematicians are ignorant of what they are doing. Much the same way as most humans in general are ignorant of what they are doing.
You usually get poetic nonsense when you ask about it.
'It's the language of universe.'
'God created the Integers'
...
<insert your favourite "mathematics is ..."poem here>
etc.

These responses are artistic, not scientific.
They stink either of a lack of understanding, or scientific defeat.

Professor_Fink wrote:

Basically, mathematics is a structure built upon formal logic with the addition of certain specific axioms. The existance of hobbits and goblins are not one of it's axioms, and so, no, Tolkein would not be counted as a mathematician.

The idea that Tolkien was a mathematician was a legally constructed conceit based on Son Goku's definition which permitted Tolkien to be a mathematician.
It was a reductio of his definition, which worked quite well albeit a little childish.

mathematics is a structure built upon formal logic

What is formal logic?
This is the study of the rules for how the mind [correctly] judges facts *apriori.
Categorical, hypothetical and disjunctive judgements comprise the fundamentals of formal logic.
It is the study of the mind's faculty of syllogisms.

Son Goku is right in saying that algebra is a study of relations with in a set together with some operators. Addition is not necessarily one of these.

Also the fact that physics is described by mathematics is largely influenced that nature was around long before mathematics was developed. Mathematics hasd grown out of the attempt of humans to describe the world around them quantatively. Sure, it has gone beyond that now, but they were its roots.

Algebra, for centuries, got on quite well without the need for the concept of a set which is a later mathematical construct. Sets are a very useful basic mathematical tool for constructing more complex concepts. They are contents of the formal discipline, not the discipline itself.
Category theory is a similar foundation without sets, but I'm not an advocate. It just demonstrates that mathematics is not simply the study of sets and operations, although sets and operations are how* mathematics [as the construction of concepts] is studied for the most part.

The question was not about the origins of the study of mathematics, that's a history question.

Finally:
Mathematics is a branch of philosophy and always was, so all mathmaticians are also philophers. There was once a time before simple minded specialists could make a living by impressing each other with how much they know about just one thing, and how ignorant everybody else is who doesn't know about that one thing.

Professor_Fink

Apperception wrote:

However, the relation of my academic titles to this thread, is itself academic, and cannot shed any light on the subject under discussion, unless there are readers simple minded enough to fall prey to some psychologically fallacious position like an appeal to fashionable authority, or some snobby version of common sense. We'd be better off without them anyway.

Well, here in lies the problem. You say that mathematicians do not understand what mathematics is, but then who defines what mathematics is? How does any other than someone who studies mathematics understand the subject sufficiently to offer an authorative definition? And anyone who has studied mathematics in sufficient depth can reasonably be called a mathematician.

The trouble is that there are a lot of weak definitions of mathematics, and so it is unlikely that any of us would agree on some set of words to describe definitively what mathematics is. Mathematics is defined wholely and solely by its axioms.

Apperception wrote:

Yes. I do believe that a lot of mathematicians are ignorant of what they are doing. Much the same way as most humans in general are ignorant of what they are doing.
You usually get poetic nonsense when you ask about it.
'It's the language of universe.'
'God created the Integers'
...
<insert your favourite "mathematics is ..."poem here>
etc.

These responses are artistic, not scientific.
They stink either of a lack of understanding, or scientific defeat.

The responses you list are generally aimed at a very wide audience, and are clearly not meant to be definitive. Indeed I believe the second is the title of a popular science book by Stephen Hawking. Additionally, it's a metaphore, Hawking isn't actually claiming that some devine being created mathematics.

Apperception wrote:

The idea that Tolkien was a mathematician was a legally constructed conceit based on Son Goku's definition which permitted Tolkien to be a mathematician.
It was a reductio of his definition, which worked quite well albeit a little childish.

Again, I have to say that this is because it is a very loose definition, and no doubt intended to be.

Apperception wrote:

What is formal logic?
...
It is the study of the mind's faculty of syllogisms.

Hmmm.... I'm not sure I'd agree with that. It's a set of axioms for how statements can be evaluated. Intrestingly, classical logic does not hold true in this universe, and is only an approximation to nature.

Apperception wrote:

Algebra, for centuries, got on quite well without the need for the concept of a set which is a later mathematical construct. Sets are a very useful basic mathematical tool for constructing more complex concepts. They are contents of the formal discipline, not the discipline itself.
Category theory is a similar foundation without sets, but I'm not an advocate. It just demonstrates that mathematics is not simply the study of sets and operations, although sets and operations are how* mathematics [as the construction of concepts] is studied for the most part.

The definition of an algebra is essentially a set U together with a collection of operators on U.

When the idea of a set was first formalised is completely irrelevant. The natural numbers are a set, the integers are a set, differential one-forms are a set, and categories are sets with additional structure (as are 3 elephants and a hamster, and if you could define some operations on them, and keep the set closed, you'd have your self an algebra).

Apperception wrote:

The question was not about the origins of the study of mathematics, that's a history question.

Actually it was. You asked why mathematics can be used to describe the universe, and that is intrinsically linked to the history and evolution of mathematics.

Apperception wrote:

Finally:
Mathematics is a branch of philosophy and always was, so all mathmaticians are also philophers. There was once a time before simple minded specialists could make a living by impressing each other with how much they know about just one thing, and how ignorant everybody else is who doesn't know about that one thing.

The time where one person could know all of mathematics has long passed. To insult mathematicians by calling them 'simple minded specialists' for not studying philosophy is both rude and naive. You also seem to have a totally warped vision of what professional mathematicians do for there salary.

Apperception

Son Goku wrote:

Do you equate an understanding of the underlying conceptual framework with an ability to express the framework in a philosophically sound way?

Yes. However, philosophical soundness is harder to establish because it's dependent on a potential rats nest of ambiguity. This allows students of one discipline license to mock the lack of clarity of those in another.

The question is not a mathematicalone, but one about the philosophy of mathematics. You can quite easily spend many years studying and teaching mathematical processes and constructions without paying any attention to the philosophy of what is going on.

For instance does my inability to give a definition of geometry that doesn't use modern mathematical terminology indicate some deficiency in my understanding of it?

What would you be understanding?
Is it the way the modern terminology is used, or is it what it actually means?
If it only means something in relation to functions, operators or sets or even just symbols, then it has no intentionality towards nature in itself, and any correspondence with it would be a mere accident.
The fact is that we CAN use mathematics to formally describe the way the universe appears to us. However, that does not mean that the 'form' of the universe itself is necessarily mathematical, just that it appears to us to be mathematical formulated.
Now this says something about how things appear to us, not about the universe.

What kind of definitions do you want? I suspect you want something that basically amounts to explaining the underlying similarities of the mental objects considered when one is doing geometry, that is independant of the current terminological basis.

Pretty much this is it. The thing is that geometry is a very special mental object insofar as it is necessarily given wholly (as the set* of rules for the construction of spatial objects), when any object is thought.
Human beings could never experience a part of the universe that does not conform to Euclidian geometry, because the mind creates Euclidian space as part of experience. We would not recognise objects at all, if the local geometry did not allow the mind to create the 'whole' object out of what is sensed.

This is why rules like 'two lines can't enclose a space' are universally valid for human experience.
To anticipate the usual objection : Yes, theoretically possible spatial topographies might allow this to be true, but they are not what is created by the mind when we experience objects [for no objects therein would be recognisable because they would be determinable only in parts], they are fancies, some of them useful, some less so.

Note: This does not preclude such an existence, just that we could never experience it.

How do algebras of functions work in this definition? Or algebras of operators?

These ideas are wholly formal and sometimes can leave the realms of meaning (intenionality towards nature) altogether, depending on the circumstances.
The problem usually arises when ontological status is granted to what are merely symbolic conveniences.

The 'content' of an algebra can be anything quantifiable, but it is relatively important. If the content has some intrinsic meaning towards nature [trigonometric proportionality], then it can be said to be philosophical.

It can also be arbitary (and more fun) like some of the higher dimensional speculations in string theory, but these things have intentionality towards nature only by academic dogma, not through demonstrated applicability towards experience.

A counter-example might be:
http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/PureTime/

This is because Hamilton knew these things, and understood what the intension of algebra is.

Professor_Fink

Apperception wrote:

It can also be arbitary (and more fun) like some of the higher dimensional speculations in string theory, but these things have intentionality towards nature only by academic dogma, not through demonstrated applicability towards experience.

Ok, let me stop you right there! The mathematics of string theory are not unique to string theory. Differential geometry is used to describe the string, with some metric for the background creating an induced metric on the string. Commutation relations are then used to impose quantization on the field, just as in any quantum theory. The mathematics used are exactly those of general relativity and quantum field theory.

Also string theory is substantially less arbitrary than you are implying.

Son Goku

Mathematics is a branch of philosophy and always was, so all mathematicians are also philophers. There was once a time before simple minded specialists could make a living by impressing each other with how much they know about just one thing, and how ignorant everybody else is who doesn't know about that one thing.

This allows students of one discipline license to mock the lack of clarity of those in another.

Are you saying that the different disciplines of maths all make fun of each other or that students might do so (e.g., Topologists mock Algebraists, e.t.c.) Or are you saying something different?

Yes. However, philosophical soundness is harder to establish because it's dependent on a potential rats nest of ambiguity.

Would that not exclude people like Srinivāsa Rāmānujan, who had without a doubt had a tremendous and deep understanding of mathematics, but never really learnt philosophy or its style of argumentation?

This is why rules like 'two lines can't enclose a space' are universally valid for human experience.
To anticipate the usual objection : Yes, theoretically possible spatial topographies might allow this to be true, but they are not what is created by the mind when we experience objects [for no objects therein would be recognisable because they would be determinable only in parts], they are fancies, some of them useful, some less so.

Non-Euclidean Geometries are fancies? In the real world spacetime is Non-Euclidean. You're making very unusual objections. Non-Euclidean Geometries just take a while to get used to, but then you can think about them fairly easily.

With regard to experiencing them, we experience Non-Euclidean geometries all the time.

You seem to be restricting mathematics to solely that which springs to humans initially.

These ideas are wholly formal and sometimes can leave the realms of meaning (intentionality towards nature) altogether, depending on the circumstances.
The problem usually arises when ontological status is granted to what are merely symbolic conveniences.

Let me ask you this:
In curved spacetimes describing the dynamics of particles requires algebras of operator valued distributions. This algebras have a topology of the space of all "uncountably infinite dimension"-tuples (Aleph-1-tuples) of Real numbers Cartesian producted with itself an uncountably infinite amount of times. Such a thing is wholly formal and yet it really helps you understand how particles behave in curved spacetimes. And eventually you learn to understand particle dynamics in this way. What is your problem with such constructions?

Apperception

Son Goku wrote:

Are you saying that the different disciplines of maths all make fun of each other or that students might do so (e.g., Topologists mock Algebraists, e.t.c.) Or are you saying something different?

I'm saying that students of mathematics might mock philosophy students because they think the subject is ambiguous (and in many cases this is true).

Would that not exclude people like Srinivāsa Rāmānujan, who had without a doubt had a tremendous and deep understanding of mathematics, but never really learnt philosophy or its style of argumentation?

Interesting character Ramanujan. I can't know what he understood, but at a guess I'd say he understood perfectly what I'm talking about and wouldn't disagree at all. None of what I'm saying is contradicted by any mathematical success story, in fact it's augmented by it insofar as it turns out to be universally valid.

Non-Euclidean Geometries are fancies? In the real world spacetime is Non-Euclidean. You're making very unusual objections. Non-Euclidean Geometries just take a while to get used to, but then you can think about them fairly easily.

With regard to experiencing them, we experience Non-Euclidean geometries all the time.

It always comes to this

Yes. Non-Euclidean geometries are fancies, or if they aren't, we cannot know about it.
SpaceTime is not non-Euclidean *locally, it's Euclidean, as are all geometries
Humans cannot have any non-local experience.

That bears repeating:
SpaceTime is not non-Euclidean *locally, it's Euclidean, as are all geometries
Humans cannot have any non-local experience.

Furthermore, it's described by incremental application of angled Euclidean spaces and necessarily so, because Euclidean spatial representation is the only one that makes sense* to human beings.

Euclidean space is the space of human sensation. i.e. It's the space that the human faculty of sensation creates when presenting objects of experience.

Half a house is not a house. If space did not allow for the representation of the 'whole' object to be grasped in a single act, then it would not be recognisable as an object. It must be possible to think each and every part of an object in any order* (top->bottom, bottom->top) before the object can be cognised. It must also be possible to think both sides of the boundary of an object, before the object can be determined.

Just because you can think about a non-Euclidean space doesn't grant it any ontological status. You can easily think a unicorn, but that doesn't make it real, or unreal either.

Non-Euclidean spaces are useful mathematical tools, but they are not the space of our experience.

You seem to be restricting mathematics to solely that which springs to humans initially.

Im restricting mathematics to the construction of concepts and stripping it of all claims to absolute ontological status. Technically this is necessary because of the subjective nature of experience itself.

You cannot remove the observer, ever.

Let me ask you this:
In curved spacetimes describing the dynamics of particles requires algebras of operator valued distributions. This algebras have a topology of the space of all "uncountably infinite dimension"-tuples (Aleph-1-tuples) of Real numbers Cartesian producted with itself an uncountably infinite amount of times. Such a thing is wholly formal and yet it really helps you understand how particles behave in curved spacetimes. And eventually you learn to understand particle dynamics in this way. What is your problem with such constructions?

I have no problem with such constructions whatsoever.
What gave you the impression I had?

These concepts are very useful for describing the way nature appears, but that does not mean that nature is actually that way, or that the concepts [as constructed by us] actually exist by themselves somewhere outside of a faculty of conception.

Apperception

Professor_Fink wrote:

Ok, let me stop you right there! The mathematics of string theory are not unique to string theory. Differential geometry is used to describe the string, with some metric for the background creating an induced metric on the string. Commutation relations are then used to impose quantization on the field, just as in any quantum theory. The mathematics used are exactly those of general relativity and quantum field theory.

Also string theory is substantially less arbitrary than you are implying.

If I said string theory is all faeries at the bottom of the garden, no mathematician or string theorist would be able to prove me wrong.
Do you wonder why Mr. Ed. "God of Strings" Witten now studies phenomenology?
- Not an appeal to his celebrity authority, merely indicative.
http://www.sns.ias.edu/~witten/papers/Unravelling.pdf

You can create all the Calabi-Yau manifolds you like, but you will never be able to represent an object located in such a space in intuition, or be able to perceive it as a complete object if it were given in a sensation which included a Calabi-Yau style spatial framework.

However, pencil in hand, you can construct such complex, imaginative and logically possible conceptions, but what then have you done?

Professor_Fink

Apperception wrote:

If I said string theory is all faeries at the bottom of the garden, no mathematician or string theorist would be able to prove me wrong.
Do you wonder why Mr. Ed. "God of Strings" Witten now studies phenomenology?
- Not an appeal to his celebrity authority, merely indicative.

String theory is a proposed physical theory. In order to be considered physics it must make verifiable predictions, and must be falsifiable. At present it doesn't. Hence phenomenology. You seem to be confusing physics and mathematics here.

This is all about string theory's status as physics, not mathematics.

Professor_Fink

Apperception wrote:

It always comes to this

Yes. Non-Euclidean geometries are fancies, or if they aren't, we cannot know about it.
SpaceTime is not non-Euclidean *locally, it's Euclidean, as are all geometries
Humans cannot have any non-local experience.

That bears repeating:
SpaceTime is not non-Euclidean *locally, it's Euclidean, as are all geometries
Humans cannot have any non-local experience.

Furthermore, it's described by incremental application of angled Euclidean spaces and necessarily so, because Euclidean spatial representation is the only one that makes sense* to human beings.

Euclidean space is the space of human sensation. i.e. It's the space that the human faculty of sensation creates when presenting objects of experience.

Ok, this is blatently false. Space is only Euclidean in the infinitessimal limit, and humans are extended objects. As long as the curvature of space time is substantial on the scale of a human body, we can experience non-Euclidean space.

An example of this is what would happen if you fell towards a neutron star. As you approach the surface, the curvature along your body would quite literally rip you in half (actually probably into more than two pieces).

Apperception wrote:

Im restricting mathematics to the construction of concepts and stripping it of all claims to absolute ontological status. Technically this is necessary because of the subjective nature of experience itself.

You cannot remove the observer, ever.

Now here I have a problem. You ask people whether they really understand what mathematics is, and then pick a controversial definition and present it as the truth.

There is no necessary requirement on the ontological status of mathematical constructs. Mathematics need not relate to the observable world to be considered mathematics. If you chose to impose this restriction (presumably to circumvent Godel), then you are not talking about what we call mathematics (1), you are talking about a subset of mathematics, which you have chosen, rather inconviently, to also call mathematics (2).

So when you ask do mathematicians know what mathematics is, and you get a definition of (1), it is because you have asked a ridiculously ambigous question. It's like the spelling bee in the Simpsons where a kid is asked to spell (whether/weather) and asks the judge to use it in a sentence. The reply is "I don't know whether the weather will be ....".

And I'd like to see a proof that you can never remove the observer.

Apperception

Professor_Fink wrote:

String theory is a proposed physical theory. In order to be considered physics it must make verifiable predictions, and must be falsifiable. At present it doesn't. Hence phenomenology. You seem to be confusing physics and mathematics here.

This is all about string theory's status as physics, not mathematics.

I was giving string theory as an example of where mathematical concepts [as logically possible constructions] have lost all intentionality towards nature, and any correspondence to it will be accidental (notice the way they are groping in the dark without any guiding principle).
The quantitive aspect (content) of experience has been removed in these theories, and only the formality remains.

It is creation, not discovery.

Professor_Fink

Apperception wrote:

I was giving string theory as an example of where mathematical concepts [as logically possible constructions] have lost all intentionality towards nature, and any correspondence to it will be accidental (notice the way they
are groping in the dark without any guiding principle).
The quantitive aspect (content) of experience has been removed in these theories, and only the formality remains.

It is creation, not discovery.

That is quite simply not true. The guiding principal behind the evolution of strings is the minimisation of action. This is exactly what we do with classical particles, quantum particles, quantum fields, particles in general relativity, etc. The problem is mainly that there are many ways to fold up the extra dimensions, and they currently all appear to be equally valid.

It's not fair to label it as creation either, since the number of dimensions is dictated by the rotational symmetry groups for massive and massless particles respectively.

Now, while string theory may not be an accurate description of reality, string theorists do at least know where they want to get to with string theory.

Apperception

Professor_Fink wrote:

Ok, this is blatently false. Space is only Euclidean in the infinitessimal limit, and humans are extended objects. As long as the curvature of space time is substantial on the scale of a human body, we can experience non-Euclidean space.

Good objection since it helps to clarify 'experience' and 'space'.

Your assumption is that the 'space' I am describing is some sort of object that human bodies 'experience'.

This is not the way our intuition of space is generated. The objects of our experience are all presented in a unified spatial field by the mind (constructed by the thalmocortical signalling loop, from signals of varying somatosensory origins).
This field is generated subjectively and contains only representations not actual objects themselves.
The spatial framework is apriori and necessarily Euclidean for the purposes of complete object creation. If it were not so, then we would not cognise the objects at all. A good example are neurologically impaired people, that have no concept of left and so cannot construct objects either in intuition, or on paper that contain a left side. (They draw half houses, but think them whole).

Furthermore, the spatial [conceptual scaffolding] framework that is constructed by the mind / brain, exists outside 'empricial space' as describable by an observer, has no extension outside the subject.
Because it has no observer determinable extension in 'real space', any curvature of said 'real space' will not affect the subjective representation.

In this sense the space that is created is fully subjective.

Your current understanding of space, seems traditionally materialistic and presumably entails notions like the 'fabric of space' as if it were something that would remain if all the objects of the universe were removed therefrom.

Of this space, we can know nothing at all since we can only formally investigate the space of which we have immediate experience, which is the space created by the mind as the vehicle of all other conceptual constructions.

That this space is analogous of an outer ontologically more valid existence is duly granted, but formal investigation of this outer space is excluded as impossible since we can only formally investigate whatever we have immediate intuition of, not anything that is mediated by a faculty of sensation, for the form that the faculty of sensation takes must first be investigated exhaustively...

Now the usual cry to this is something about Einsteins on-intuitive SpaceTime, but that uses the forms of subjectively given Euclidean space to make predictions about the behaviour of phenomena.

It's no mistake that he worked it out by using his intuition and a pencil.
BTW: Einstein was well versed in this theory, but that's up to the physics and philosophy historians to tell.

Note:
Phenomena [the phenotype of noumena] are appearances in the subjectively given Euclidean Space of experience.
Noumena are the postulated 'real objects' in space outside sensory experience.

And I'd like to see a proof that you can never remove the observer.

Can you even show it's possible to remove the observer?

Apperception

Professor_Fink wrote:

The problem is mainly that there are many ways to fold up the extra dimensions, and they currently all appear to be equally valid.

The reason for this is the synthetic nature of mathematics. There will be any number of ways to describe how to construct any concept. Arithmetic is the perfect example. 7 + 5 , 144/12 etc...

It's not fair to label it as creation either, since the number of dimensions is dictated by the rotational symmetry groups for massive and massless particles respectively.

Ok. It may be a little disparaging of me to label the whole thing as creation, but it opens a debate, which is good so long as it's an intelligent one.

The potential merits of the study of strings is slightly off this topic but can be fun and usually gets bogged down (usually over drinks) once interpretations of QM are brought into it, which is were it enivitably ends up

Son Goku

Apperception wrote:

I'm saying that students of mathematics might mock philosophy students because they think the subject is ambiguous (and in many cases this is true).

This is what I suspected the thread was about. Students of maths, like students in any subject, can say silly things. People tend to appreciate their own interests more than the interests of others. This is nothing remarkable and not a cause for criticising mathematics.

Interesting character Ramanujan. I can't know what he understood, but at a guess I'd say he understood perfectly what I'm talking about and wouldn't disagree at all.

Based on what?

None of what I'm saying is contradicted by any mathematical success story, in fact it's augmented by it insofar as it turns out to be universally valid.

What? Ramanujan couldn't even write good proofs, let alone make rigorous philosophical arguments and yet there was probably not a mathematician this Century, in his area, with anywhere near his ability.
I think this counters your statement that in order to understand the framework of maths deeply you need to be able to express the framework in a philosophically sound way.

Yes. Non-Euclidean geometries are fancies, or if they aren't, we cannot know about it.

Footballs are Non-Euclidean. I don't understand what is so "off-limits" about the concept of Non-Euclidean geometry. Besides Euclidean geometry is infinite in extent, no human has ever experienced an infinite amount of space. What makes Euclidean Geometry so special?

SpaceTime is not non-Euclidean *locally, it's Euclidean, as are all geometries
Humans cannot have any non-local experience.

So? What is all this talk about experience?
If spacetime behaves in a manner so similar to Minkowskian or more general Lorentzian spaces that we can't tell the difference at the classical scale, then what’s the problem?
Sure, spacetime mightn't even be Euclidean on very small scales. I really don't understand what makes Euclidean geometry so great. It's just as much a model that mightn't have anything to do with reality as anything else.

Furthermore, it's described by incremental application of angled Euclidean spaces and necessarily so, because Euclidean spatial representation is the only one that makes sense* to human beings.

That isn't true. A Non-Euclidean manifold that can be pictured in your head is a football. Compare this with R^4, a Euclidean space which can't be pictured.

Half a house is not a house. If space did not allow for the representation of the 'whole' object to be grasped in a single act, then it would not be recognisable as an object. It must be possible to think each and every part of an object in any order* (top->bottom, bottom->top) before the object can be cognised. It must also be possible to think both sides of the boundary of an object, before the object can be determined.

Just because you can think about a non-Euclidean space doesn't grant it any ontological status. You can easily think a unicorn, but that doesn't make it real, or unreal either.

Just because "you can think about a Euclidean space doesn't grant it any ontological status. You can easily think a unicorn, but that doesn't make it real, or unreal either". Again what I don't understand is Euclidean space being made special.

You're being very vague. If Non-Euclidean geometry describes spacetime better than Euclidean geometry doesn't it deserve consideration over it. Regardless what our experience is.

Besides earlier you were saying that mathematics should have intentionality towards nature. Non-Euclidean geometry attempts to describe nature and one can think about it and yet you still rule it out.
Why?

Im restricting mathematics to the construction of concepts and stripping it of all claims to absolute ontological status. Technically this is necessary because of the subjective nature of experience itself.

You cannot remove the observer, ever.

Is this a thread about mathematics as it appears in theoretical physics or pure mathematics. You seem to be switching between the two without indicating that you are doing so. The String Theory stuff makes no sense if your talking about pure mathematics.

These concepts are very useful for describing the way nature appears, but that does not mean that nature is actually that way, or that the concepts [as constructed by us] actually exist by themselves somewhere outside of a faculty of conception.

So?
You started of this thread about mathematics, now it's about the world not actually being as theoretical physics describes it. In fact the group you've been criticising in your last two posts are theoretical physicists and not mathematicians.

Which subject are you discussing?

Professor_Fink

Apperception wrote:

Your assumption is that the 'space' I am describing is some sort of object that human bodies 'experience'.

I'm not saying space is an object, but we most definitely experience the effects of curved space.

Apperception wrote:

This field is generated subjectively and contains only representations not actual objects themselves.
The spatial framework is apriori and necessarily Euclidean for the purposes of complete object creation.

That is a completely unsupported claim. Why do you believe any representation must be necessarily Euclidean?

Apperception wrote:

Your current understanding of space, seems traditionally materialistic and presumably entails notions like the 'fabric of space' as if it were something that would remain if all the objects of the universe were removed therefrom.

That would be an extremely inaccurate description of my 'understanding' of space. Again, you need to make it clear what you are talking about when you say space, as there are several things which go by that name. If you mean something constructed only in our minds, then you need to make that explicit, and I still see no reason for it to be exactly Euclidean. Clearly that space could not exist without a brain (or something) to represent it, etc.

If you are refering to a manifold, responsible for gravity, etc., then you need energy, and if you have energy then you have something, and not nothing. So no, space cannot exist on it's own, and it makes no sense to talk of this type of space except by reference to it's effect on particles etc.

Apperception wrote:

Now the usual cry to this is something about Einsteins on-intuitive SpaceTime, but that uses the forms of subjectively given Euclidean space to make predictions about the behaviour of phenomena.

It's no mistake that he worked it out by using his intuition and a pencil.
BTW: Einstein was well versed in this theory, but that's up to the physics and philosophy historians to tell.

Actually relativity did not come from intuition, but rather an inconsistency between the understanding of electromagnetism and the translation rules between observers at the time.

Apperception wrote:

Can you even show it's possible to remove the observer?

You asserted it was impossible. I asked for a proof of that statement, and all you can offer in response is to ask if I can prove it possible. This is no proof at all. The fact that I do not offer you a proof, does in no way imply that a proof, one way or the other, is impossible.

If you are going to make assertions like this, you had damn well better be able to back them up if you expect your arguement to be taken seriously.

Apperception

Son Goku wrote:

This is what I suspected the thread was about. Students of maths, like students in any subject, can say silly things. People tend to appreciate their own interests more than the interests of others. This is nothing remarkable and not a cause for criticising mathematics.

It's not just students and not just mathematicians and not just limited to cross discipline disputes. Physicists from one are look down on those from another. Maybe it's an academic necessity, I don't know, but it's counterproductive.

Based on what?

Nothing, it was a guess. I can't know what he thought. I didn't even bring him up Maybe he had no understanding at all, but his cultural background is steeped in idealistic interpretation of reality anyway, so there is some hope that he did have a deep understanding. There's no way for us to know.

What? Ramanujan couldn't even write good proofs, let alone make rigorous philosophical arguments and yet there was probably not a mathematician this Century, in his area, with anywhere near his ability.
I think this counters your statement that in order to understand the framework of maths deeply you need to be able to express the framework in a philosophically sound way.

There's no was to know how he did the things he did, just the same as there's no was to know how an autistic savant can reel off 15 digit primes.
Just because they can do it doesn't mean they understand the meaning of what they are doing.

Footballs are Non-Euclidean. I don't understand what is so "off-limits" about the concept of Non-Euclidean geometry. Besides Euclidean geometry is infinite in extent, no human has ever experienced an infinite amount of space. What makes Euclidean Geometry so special?

The space of experience is infinite and unified.
When you think of any object, the space it is in is infinite.
If you think of a finite space, you must represent it to yourself as contained with a larger infinite Euclidean space.

Footballs are just objects that you present to yourself in space like any other object.

If you create a football shaped space mathematically, you must present it to yourself [by imagination] in intuition as contained within a larger 'background' infinite Euclidiean space.

Just because "you can think about a Euclidean space doesn't grant it any ontological status. You can easily think a unicorn, but that doesn't make it real, or unreal either".

Again what I don't understand is Euclidean space being made special.

You're being very vague. If Non-Euclidean geometry describes spacetime better than Euclidean geometry doesn't it deserve consideration over it. Regardless what our experience is.

Ok. I'm not denying or asserting anything about the nature of 'real' , or noumenological space. Whether or not it's Euclidean in itself is not decidable.
Nothing about it is ruled out.

This thread is about the nature of the content of mathematical investigation.
That the content has intentionality towards nature would presumably be the claim of any mathematician, otherwise what would be the point of it at all?

Is this a thread about mathematics as it appears in theoretical physics or pure mathematics. You seem to be switching between the two without indicating that you are doing so. The String Theory stuff makes no sense if your talking about pure mathematics.

Agreed, strings were possibly a poorly chosen digression and merely an attempt to show some misdirected intentionality.

I suppose the thesis that Mathematics "is the study of the construction of concepts" rather than the study of something else can end up wandering off the point from time to time, but that's an unfortunate part of a limited attention span.

Professor_Fink

Apperception wrote:

The reason for this is the synthetic nature of mathematics. There will be any number of ways to describe how to construct any concept. Arithmetic is the perfect example. 7 + 5 , 144/12 etc...

No, that is not the reason. GR and QFT both make extremely accurate predictions, and rely on essentially the same mathematics.

String theory is simply not sufficiently well understood at this point to be verified or falsified. The idea that we have not confirmed its status because of some flaw in mathematics is frankly laughable, or would be if it wasn't so seriously misguided.

And different compactifications are NOT different representations of the same concept.

What possible evidence can you offer in support of the assertion that lack of progress in string theory is linked to some flaw in mathematics?

Apperception wrote:

Ok. It may be a little disparaging of me to label the whole thing as creation, but it opens a debate, which is good so long as it's an intelligent one.

The potential merits of the study of strings is slightly off this topic but can be fun and usually gets bogged down (usually over drinks) once interpretations of QM are brought into it, which is were it enivitably ends up

Interpretations of QM are completely seperate from the behaviour of quantum mechanics, and in this case from string theory. Interpretations are just that! Interpretations! They are all based on the same mathematical framework, and necessarily all predict the same phenomena (unless one is a misinterpretation of quantum mechanics). The make absolutely no impact on the validity of string theory.

Mathematicians and physicist try to leave the discussion until we're sober, since it isn't safe to drink and derive! ;-) Couldn't resist it!

Son Goku

Apperception wrote:

It's not just students and not just mathematicians and not just limited to cross discipline disputes. Physicists from one are look down on those from another. Maybe it's an academic necessity, I don't know, but it's counterproductive.

So? Similarly there are philosophers who think relativity is nonsense because it isn't philosophically sound. It's just somebody thinking their subject is the best. There's assholes in every academic discipline. It doesn't really say anything special about Mathematics.

(Although physicists in one area looking down on physicists in another is pretty rare, maybe you mean physicists who look down on other disciplines).

Apperception wrote:

There's no was to know how he did the things he did, just the same as there's no was to know how an autistic savant can reel off 15 digit primes.
Just because they can do it doesn't mean they understand the meaning of what they are doing.

That is weak. Ramanujan wasn't educated, but he was no savant, you're just dodging the point. He understood number theory to a great degree, yet couldn't argue about it philosophically. By these facts you rule out one of the greatest geniuses of the century as somebody who didn't understand his own work. Claiming any similarity to savants is just lazy.

The space of experience is infinite and unified.
When you think of any object, the space it is in is infinite.

Really? I don't think of space as infinite. And what does unified mean?

Footballs are just objects that you present to yourself in space like any other object.
If you create a football shaped space mathematically, you must present it to yourself [by imagination] in intuition as contained within a larger 'background' infinite Euclidiean space.

Yeah, but they're still Non-Euclidean, regardless of what they're in.

Ok. I'm not denying or asserting anything about the nature of 'real' , or noumenological space. Whether or not it's Euclidean in itself is not decidable.
Nothing about it is ruled out.

This thread is about the nature of the content of mathematical investigation.
That the content has intentionality towards nature would presumably be the claim of any mathematician, otherwise what would be the point of it at all?

Then you presume wrong. Mathematicians mostly just do maths, they don't really claim anything about it. In fact I think you'd find most wouldn't even know what you meant by that.
As to the point of it all, I don't think you understand why people do pure maths. It's the same as asking what the point of a Picasso is.

It's theoretical physicists who use maths with an intentionality towards nature. I think you're mixing up the two groups.

babytooth

is maths not a language, nothing more or less, that describes, in the most base of levels, the workings of the world.

It philosophizes, it augments and allows for improvisational thinking and conceptual manipulation in an attempt to model what one needs to be broken down into understandable packages.

Geometry, is the art/ability to use maths to enable visualization on concepts with regard to their individual relationships within multi-dimensional elements, ie: 3d or 4d......this relationship is mutli-directional, as in maths drives visualization as visualization drives maths, take the concept of log as an example, which came about through the attempt to derive an equation that enabled its line to be drawn in the sands on ancient Greece.


Algebra, this is probably easiest described as the puzzle-breaker. It is the skills and rules to take in, manipulate and take out information from an information set whilst maintaining the integrity of the input information and upholding common "truths" and axioms.


All in all, I would describe maths as the base language, sort of an open-source code....an ability to describe as best and as truly as possible what they see about them, what the conceptualize and what patterns they see through out the world.

Someone mentioned string theory, which is described by maths, or which, should i more accurately say, attempts to describe its phenomena based on the underlying thinking....
Maths is not perfect, as the stock market shows, but it does allow us some degree of simplification, of conceptualization and derivation in order to aid our understanding.

excuse the long-windedness and mis-spelling.

Fremen

Ok, a lot of this stuff is pretty dense, and it's going to take me a while to read and internalise it all. While I'm doing that, I thought I might just chime in on the non-euclidian geomety issue.

Human beings could never experience a part of the universe that does not conform to Euclidian geometry, because the mind creates Euclidian space as part of experience. We would not recognise objects at all, if the local geometry did not allow the mind to create the 'whole' object out of what is sensed.

This is why rules like 'two lines can't enclose a space' are universally valid for human experience.

Anyone who played with a globe as a kid will have direct experience with a totally comprehensible non-euclidian geometry.
Lines are great circles around the globe, and the usual axioms hold, aside from the one about parallel lines (I think it's called the fifth postulate IIRC).


For what it's worth, I've always (well, since I gave it some thought) considered mathematics to be defined as follows:
"Mathematics is a way of thinking about things characterised by rigour and abstraction."

Of course, you can pick holes in this all day. Mathematics is only as rigorous as peer review can make it. What I hand up as homework is mathematics, but it's not necessarily rigorous.
And so on...