Mathematics as an a priori field of inquiry vs. the idea of the input of the mathemat

[Wilde Mutton]

Which is it anyway, and to what line? Certain "truths" seem obvious but are in fact axioms that can be defied through other mathematical universes (and there are some, challenging the Euclidian way of the world [and the accuracy of the "Euclidian glasses"]). I at least used to be a Platonist to some degree - mathematics to me then was something that was formulated from objects already there, and every formula was to be derived from another one, all formulas eventually backing themselves up on a few basic qualities of THE mathematical universe (Euclidian, that is). It was only recently that I climbed a bit downwards from that ivory tower and chanced upon the remark that mathematics was something partially created by the mathematician. If so, to what degree, and heck, why not completely? While I´m at it, the connection between reality and mathematics might be dealt with as something inexorably linked with the topic. What is this all about? NB: I do not know anything about mathematics. Take no offence. Enlighten me...

[crofbe]

afaik, Godel's incompleteness theorem basically destroyed the concept of absolute mathematical formalism (which Russell and Whitehead tried to do in Principia Mathematica following the urgings of Hibert). I don't know First order logic very well though, so I don't really understand the significance of the theorem .

A higher approach to mathematics (not very high, just above logic), though, shows that there are deep underlying structures to the concepts we define (yes it's a circular argument). This was basically Kant's question though, that how are all these inevitabilities possible. Why is it that the 2 transcendental numbers e and pi occur everywhere, and in some cases, are used in tandem? Why are many relations linear?

So, I don't think there are contradictory elements in mathematics (or at least not famous or easy to explain ones), and the input of the mathematician (rigour and more rigour) has only gone to show these a priori facts. Input says 1+1=2, but in the a priori case we know this isn't pure arithmetic, because like you said, it's abstract.

There are definitely a lot of holes, and re: axioms - you can check Zorn's Lemma, a.k.a. the Axiom of Choice. It sucks ass and it's probably useless. But it was proved to be incomplete w.r.t. the ZF set axioms.

Sorry for the technical response, please correct me if I'm wrong. I'm still a bit bitter after math got way too hard and abstract :lol: Wikipedia is pretty good for giving summaries of the Axiom of Choice and Godel, and mathworld.com is a bit more technical.

[Wilde Mutton]

A higher approach to mathematics (not very high, just above logic), though, shows that there are deep underlying structures to the concepts we define (yes it's a circular argument). This was basically Kant's question though, that how are all these inevitabilities possible. Why is it that the 2 transcendental numbers e and pi occur everywhere, and in some cases, are used in tandem? Why are many relations linear? Why is it that prime numbers follow ln(x)/x asymptotically, and so elegantly? And then there's mathematical physics, which is probably some of the hardest shit in academia today.


A question then arises whether mathematics can be used in such sciences as physics what with its contradictory nature. Why does mathematics seem to help us so much when it comes to dealing with the phenomena of the "outside world"? Is mathematics helping us describe anything real, or is it a coincidence that it seems to back up what we have found out about natural laws so far and continue to find out? To put it bluntly, is mathematics nature´s language, a decoder of sorts, or are the results we obtain with it just a projection of something that is (accidentally?) in our brains and that therefore seems so "elegant" and "inevitable"?

[Wilde Mutton]

Sorry for the technical response, please correct me if I'm wrong. I'm still a bit bitter after math got way too hard and abstract :lol: Wikipedia is pretty good for giving summaries of the Axiom of Choice and Godel, and mathworld.com is a bit more technical.

I´m bitter about math in general. It´s something one wishes to excel in and in which one at first doesn´t see any logical reason for failure. If everything follows, or is, at least, a fairly unified system, unified rather logically, at that, then why on earth should it be so hard? However, it just seems to be. :rant:

[Claverhouse]

chanced upon the remark that mathematics was something partially created by the mathematician.
According to Swedenborg... Everything external to the self is merely a reflection or projection of what lies within the self. This includes every item and every landscape. I prefer not pondering this too long, since I want to retain my sanity.



If so, to what degree, and heck, why not completely? While I´m at it, the connection between reality and mathematics might be dealt with as something inexorably linked with the topic. What is this all about? NB: I do not know anything about mathematics. Take no offence. Enlighten me...
Don't you take offence; this is only addressing one of your points through my sheer ignorance. I've explained enough that all math is a closed book to me, so even when I here quote the master I have no idea of his concepts or meaning: but according to Spengler:


In his book Restivo tries to rehabilitate Spengler by exploring his thesis ‘that there are as many number-worlds as there are cultures’. To many (social) scientists mathematics is a unique mode of knowing that has produced universal orderings. Spengler, however, attacked this privileged status of mathematics as an intellectual or scholarly discipline: ‘We find an Indian, an Arabian, a Classical, a Western type of mathematical thought and, corresponding with each, a type of number - each type fundamentally peculiar and unique, an expression of a specific world-feeling, s symbol having a specific validity which is even capable of scientific definition, a principle of ordering the Become which reflects the central essence of one and only one soul, viz., the soul of that particular Culture. Consequently, there are more mathematics than one.’ (Spengler, Cited in Restivo, p8)

From this review (http://www.easst.net/review/sept1996/bogaard) of a work by the other gentleman. Plenty of other pages on Spengler and mathematics; but --- and this illustrates your thesis to a minor extent --- I am only interested in the application of his ideas, not the math he grounded them on...





Claverhouse :ph34r:



But, this one (http://www.marco-learningsystems.com/pages/sawyer/mat_hist.htm) explains one example of what he meant ( before going on to prove that he was wrong about our decline ( which he isn't, and he left plenty of lee-way time in years for events to unfold: not this will happen about 2000ad, but more it will happen sometime around 2000ad give or take 100 years; he wasn't a bloody astrologer )):



For instance, early in the book he scoffs at those who take a superficial view of history, such as the Jacobin clubs in the French Revolution, who had a cult admiring Brutus as a revolutionary. Spengler points out that Brutus was in fact "a millionaire and an exploiter who, as a supporter of the oligarchic regime, stabbed the man of the democracy amid the plaudits of the aristocratic Senate." [4] I never realized this, when I did Julius Caesar at school. According to Spengler, there was a close analogy between the history of a culture and the life of a human being. A culture was born; there followed a vigorous youth and a period of maturity; finally came an age of decadence, in which old beliefs and loyalties faded and money dominated everything. He gave the years A.D. 1800 to 2000 for this stage of our society. Without admitting the truth of his general theory, one must admit that this has considerable resemblance to the
situation to-day.


He maintained this progression, from vigour to decadence, was as inevitable as the successive changes in a person who survives from infancy to old age. Moreover, at each stage, every subject, every activity reflected both the surrounding culture and the stage that had been reached. His example of a subject depending on the civilization in which it develops -, we think of our geometry as being the same as that of the Greeks, but in fact it is quite different.It is impossible for us to understand the classical view that everything is inside the sphere of the fixed stars. A child to-day, shown the classical picture of the universe, would immediately ask, "What is outside the largest sphere?" Spengler says classical man had neither the word nor the concept of empty space.[5] Greek geometry deals with the sizes and shapes of material objects.

Modern geometry is entirely different. It may start, "Suppose there are three kinds of objects, called points, lines and planes, with the property that any two points determine a line...." and so on. It is a study of logical relationships, quite distinct from Euclid, in which you can pick up the triangle ABC and put it down on the triangle DEF. The Greek fear of the distant. Spengler says the Greeks had an acute fear of anything distant in time or space. He relates this to the existence of the Polis. the city-state in which Greeks lived. "Home for classical man was what he could see from the citadel of his state. Anything beyond this was strange, indeed hostile." [5] This feeling of dread came to attach to distance in any form. Here we may begin to think Spengler is letting his imagination run away with him. But in another place we read, "In the last years of Pericles a law was passed in Athens that threatened with the severe punishment of impeachment anyone who propagated astronomical theories." [8] Such theories seem perfectly harmless to us, but evidently upset the Greeks very much. Spengler's explanation seems as good as any.

I never cared for Pericles anyway.

[crofbe]

A question then arises whether mathematics can be used in such sciences as physics what with its contradictory nature. Why does mathematics seem to help us so much when it comes to dealing with the phenomena of the "outside world"? Is mathematics helping us describe anything real, or is it a coincidence that it seems to back up what we have found out about natural laws so far and continue to find out? To put it bluntly, is mathematics nature´s language, a decoder of sorts, or are the results we obtain with it just a projection of something that is (accidentally?) in our brains and that therefore seems so "elegant" and "inevitable"?

See, in one sense, it is a projection: take an apple, take another, and it's obvious that 1+1=2. Take the limit of continual interest, and the number 'e' is inevitable. Then slightly more application reveals that 'e' is in the strangest and unexpected of places. So we projected it onto something that had always been natural law.

We'll just have to wait for the second coming of Immanuel Kant.

[Hypnos]

[...] To put it bluntly, is mathematics nature´s language, a decoder of sorts, or are the results we obtain with it just a projection of something that is (accidentally?) in our brains and that therefore seems so "elegant" and "inevitable"?
Your question begs the result.

And rightly so.

[coldmine]

There are definitely a lot of holes, and re: axioms - you can check Zorn's Lemma, a.k.a. the Axiom of Choice. It sucks ass and it's probably useless. But it was proved to be incomplete w.r.t. the ZF set axioms.



The axiom of choice is not a hole! It's certainly a controversial axiom though. Most mathematicians accept it, because their lives are much easier if they do. A lot of set-theoretic constructions would be difficult, if not impossible, to accomplish otherwise. So in short, unless you're a set theorist, you don't think much about the axiom of choice.

[eyebyte_atWork]

... To put it bluntly, is mathematics nature´s language, a decoder of sorts, or are the results we obtain with it just a projection of something that is (accidentally?) in our brains and that therefore seems so "elegant" and "inevitable"?

Mathematics is a man made tool - a guide to knowledge- if we know nothing else we know X. It is pure deductive reasoning. Nothing more - nothing less. What it means or what it describes it a totally different question. Mathematics is true - because of the strict rules applied to it (it has to be consistant with itself and therefore any result correlated to it must also be true - again pure deduction).

As far as the Euclidean comment - this refers to geometric contructs used in mathematics. All mathematical laws must hold true in all geometries if 1.) the Geometry is valid and 2.) The law is valid. Most of the arguments around these have to do with proving these or with the new results given by "playing" with multiple geometries. What is at test is the geometries - not their meaning - that comes after the math is verified.

Mathematics can not give the answer to life, the universe, everything. (Or at least their purpose - because "why" is a human learning tool - and not an answer)

Hope this helps - if it does not... oh well.

[Hypnos]

Mathematics is a man made tool


Mathematics is true - because of the strict rules applied to it

These are contradictory. Math is the rules.

Wilde Mutton's original question is the same posed by Kant -- is math wholly synthetic or wholly a part of nature? The only counterargument to the synthetic hypothesis is that it begs the question.

[eyebyte_atWork]

Only because it proves itself. Math is rules - but we only understand what we can of these rules - and what we have discovered is man understood, man comprehended and man modified or extended. Our understanding of math is man made. Math as we know it is man made. Just because it is also true does not mean we did not create it - Math is a creation that describes reality and is not reality itself.

btw - Thanks for your input - I had to think of my answer for a minute to see if my arguement wasn't flawed.

[Hypnos]

Only because it proves itself. Math is rules - but we only understand what we can of these rules - and what we have discovered is man understood, man comprehended and man modified or extended. Our understanding of math is man made. Math as we know it is man made. Just because it is also true does not mean we did not create it - Math is a creation that describes reality and is not reality itself.
Then if space aliens came, they might have a different math that one could not show to be equivalent to ours.

In doing so, math would be false, and then not math.

I would say that our understanding of math is man-made, but math itself is metaphysical and apart from man.

[eyebyte_atWork]

Then if space aliens came, they might have a different math that one could not show to be equivalent to ours.

In doing so, math would be false, and then not math.

I would say that our understanding of math is man-made, but math itself is metaphysical and apart from man.


I think we are in agreement on this.

But

if aliens provided a verison of math contraray to ours then math would have to be expanded. Math is always true and thus has to change every so often to contain a bigger picture of truth.

Like I said I think we are in agreement. Long live Mathematics.

[Loco_Mullus_Surmuletus]

Mathematics has always existed, and always will. Man can only discover and comprehend some of its beauty, buc can never erase\change it. It exists apart from time\space variables.