I have already said how arithmetic comes about. Addition is counting forwards, subtraction is counting backwards, multiplication distributes over addition, and division is its inverse. That's how everyone apprehends arithmetic in the real world from a very young age, including you. Counting is basically the idea of order, or that one thing follows another. You learn to count by putting meaningless symbols in order, which turns these symbols into numbers.

As for geometry, there are two ways in which one can count: in a line or in a circle, which are the most fundamental concepts in geometry, although I accept that geometry from counting is somewhat tenuous.

P.S. Thanks for the ninja, Carlo.

Or maybe not. I don't know how a circle could be defined without using distance, which would make distance more fundamental.

OK, to say "many mathematicians" is foolish and tantamount to saying that "everybody does it", lol.

This is interesting concerning Pythagoras who saw, supposedly, that number or math is discovered not invented:

http://science.jrank.org/pages/10928...s-Harmony.html

Back to Plato, there are of course disclaimers but I'm not trying to copy or advocate Plato's Republic with respective to math but only to somehow delineate a modern look at this realm of mathematics which, to my mind, becomes clear but not for all. The Platonic approach to math is discussed within the philosophical and mathematical intellects and that was my purpose , to give ancient and modern consideration, but not to speak from authority.

Discussion of thinking and thoughts will inevitably lead to mathematics within the same perspective but I'll stop here .

Maths comes from the Cross. Intersecting lines.

That's what all's about.

Good question. It's in our conditioning where the secret brain police reside.


PairTheBoard

You need a Point - “that which has no part,” indicates that Euclid's definition of a point as having no width, length, or breadth, but as an indivisible location. Once you have an indivisible location then distance from that location is possible. Think about R, the Radius of a circle, it is a line (again see Euclid) that begins at a point. Point, line, distance, arc, circle, each a building block or element for the structure of geometry.

Some ideas are very powerful, and there is a profundity in that. Of course. Why that necessitates invoking a Platonic realm is unclear to me.

(Most mathematicians believe in a Platonic realm which is, moreover, unknowable. They believe that non-computable real numbers exist in this realm, and that uncountably many of them exist.)

Charlie - to understand why a platonic realm is necessary you need to first admit that numbers have a kind of existence, which is neither entirely mental nor physical.

From this you're probably thinking, well then the solution is simple - they exist from the combination of the mental and the physical. This cannot be however. If the mental is necessary for the emergence of numbers then they may as well be entirely mental. Yet this is not the case. Moreover if a combination is necessary then your ontology (list of things that exist) will need to reflect that, and there's very few ontologies in which the mental and physical are simultaneously/mutually necessary (looking at Schopenhauer).

Your own ontology, for example, is likely one where the mental emerges from the physical, through evolution and so on. It's not an ontology where the mental and physical are mutually necessary for any kind of existence.

That is Euclid's definition, but I doubt it's how the manufacturers of F1 wheels do it.

I already said ITT how I think numbers come into existence. Here it is again:

How true, but mathematicians miss so much of the practical.


......well, by Jove, now you bring up the point,' said Monty frankly, 'I'm dashed if I know just what it does signify. Wheels within wheels. Why wheels? - P.G. Wodehouse.

Wodehouse was never happier than when at Dulwich College, apparently. It must be the most beautiful school in London. They teach Greek and Latin to the nobs there, which is how Raymond Chandler learnt to write. Carry on.

Right.

So to Charlie - numbers don't exist unless they're being counted by someone. Neither does order apparently, because he equates the two.

Constructive.

You better believe it. That someone however need not be a brain, it is nature itself with its structure and interactions that is doing the counting or the equivalent to counting through processes.

So yes i am here to argue Math cannot exist without Nature. You need at least one rich enough real world for math to get started because an idea is nothing without examples of its structure and properties. How are you going to describe it or start the math game without the initial blocks of the game?

Sure you can now imagine megaverses (and their mathematics) without same physics and no intelligent life etc that can prove bs or the real theory of all possible physics that can ever exist but none of this is possible without first a universe that creates this ideas in some representation form. You can also imagine unrealizable worlds with math properties etc in your brain but the brain creates theses and represents them for you in this universe using parts of that universe (cells) . Now notice you do not need the human brain to imagine those fictitious worlds. They have an independent "existence" surely outside this world of people and life but only because of this world or someone like that rich enough that gets the game started or. The ideas come to life only with representations. There can never be a discussion without the founding example!

Surely all the math we will ever develop is probably already out "there" waiting because it exists already in nature in some way even the very abstract math that appears to have nothing to do with physics, its already properties of this world (since it can clearly crate brains that get there!). I mean if a thing like theory of everything exists it is already out there true from the first day.

If nature is indeed doing the counting Masque, then do you consider nature to be intelligent?

Definitely but not in the organized way we perceive intelligence. I mean nature can solve a problem in seconds that takes current computers millions of years to simulate etc. You put the atoms there and they interact and give you the result that the simulation will take decades to achieve.

If nature is doing the counting then our problem of the existence of numbers is resolved. The mental aspect essential to the existence of numbers is present in nature, which is intelligent/has mental capacity (to count numbers, etc.). If nature is everpresent therefore, so are numbers.

Would you characterise yourself as a 'deist' or 'pantheist' then Masque?

“The world is my country, science is my religion.” Christiaan Huygens


Simply a physicist with a western civilization cultural background.

If one goes back in time there are echos of mathematics seen differently as er example the school of Pythagoras which saw "number" as the basis or reality.

The Pythagorean school was one of the last vestiges of an inner thinking/perception which was common to ancient man but nor evident during the modern age, specifically since the 15th century.

When we speak of "abstract" mathematics this means that our math has been abstracted from other sources .It literally means that the life has been removed from the mathematical (or other, word, etc..) concept and we see a corpse of a living thought to which in ancient times we could elicit as a reality.

If we count 1,2,3,... we add one unit to the previous and there is no consideration given to the idea that the cipher "1" or "1" stroke of the pen is different than "2" ciphers or "2" strokes of the pen, |,||.

In the inner aspects of a previous thought process the "|" would be considered the unity and experienced thereof. The "||" would be considered in the aspect that the individual ciphers , written or thought properly, would not be considered a whole for each of the elements would be felt as if they are separating out into space. The living quality of the two ciphers would be evident but unfortunately the typewriter, the most abstract of machines, doesn't offer much help in this perspective.

We can go to the next element which may help.

http://wn.rsarchive.org/Lectures/GA2.../204-08-02.gif

In this image there is a closed figure to which the third process belongs and it is imagined as lying in a circle, an entity within itself.

And so in the genius of language the first is split up into the second and the third closes as a whole, entirely different that the first two entities.

The Arabic symbols, 1,2,3,.. are pure abstractions of an inner consideration which was available to ancient Man, in which the numbers, so to speak, have had the life drained from them . This is the world of number which I have poorly attempted to represent and from this one can glean how the Pythagorians ,though these abilities were mostly lost to Man , spoke of the cosmic numbers but not the abstracted numbers to which we are familiar.

In effect the numbers are within Man in a form which is experiential or perceptible and actually the problem broadens with our abstract thinking.

Solution, aphoristically is the bringing forth "imaginative" thinking in order to give life to our thoughts and in this the world reality will become evident. If we do our science in abstractions then the science is a dream world science in which entities are hypothesized from our inner dream life which cannot grasp that reality to which we believe we are seeing.

The last paragraph is more than tough but make no mistake about it, abstractions, which were necessary and appropriate, have come to the fore but must be left in the progression of the evolutionary movement of Man. the thoughts will be enlivened irrespective of the contraries which will be present.

One reason is that the intellect and reason is common to all men and not the ownership of one group or individual man no matter how reverent we are to specific individuals. The Greeks spoke of "nous" or the higher faculties as a divine "gift" but not the ownership of the low ball, only the earth. Finis.

Which is the real number one, Carlo? The number one in the natural numbers, or the number one when you count time? Those are numbers on your clock, aren't they?

I believe that Minhyong Kim, a number theorist, says philosophy of maths should be like a bank account.

One builds up credit by doing proper pure maths research. Speculation on philosophy takes money out of the account.

I suspect many people in this thread are overdrawn.

They obviously have their own respective realities nor is the abstracted number an aberration . I should have written the method like this; o,oo and then the third or globules in a circular as referenced.

I expect the point is that not only numbers but words have been abstracted from a life bearing reality which does manifest in a qualitative manner. I hesitate to use that word after observing Dennett mocking any one who works with quality.

Another way of saying this is that there is a "feeling experience" with numbers which displays their qualities. This "feeling" is not our response to the number but the activity of the number acting upon us, no solipsism here.

"We can feel the wind of the north who speaks to our hearts is a dour light of existence"

There can be a qualitative mathematics but I'm not able at present. In this thread a reference was made to the Fibonacci sequence which can be related to the growth of a plant. As the plant grows from root to stem to leaf to calyx to flower to fruit or seed a spiral growth is manifested in the plant, more like a helical movement.

The Fibonacci sequence goes 1,3,5,8,13,...with the succeeding number the additive of the previous two numbers. This is the abstracted Fibonacci placed upon a living organism which doesn't fit but thee is this helix or spiral which also has quality and so to glean the working of inner numbers the Fibonacci points to the inner number sequence which becomes a matter of metamorphosis.

In the scientific parlance we look for causes or forces which are point forces but this cannot be done with the plant which has life and we are dealing with activities external to the earth to which the plant grows.

Of course , no scientist would ever deny the work of the sun in this matter but the forming activity of the extratelluric manifests with, but not only, the sun but we, our science presently, claim all growth here in a causal manner and secondary to the inorganic, whether believed or not.

This should bring in the censor but present science states that the sun is exploding gases in space as on the earth. This is consequential , in part, to our abstractive presentations which only see the telluric, abstractly and project our beliefs out into space. there are to many avenues here and I'll stop.

They obviously have different properties, but what property do they have in common that makes them both the number one?

The answer is that one is the unique number which, when multiplied by any number x, gives x. This property is surely the essence of the number one.

I'd like to add that when I speak of the ancient scientist and his abilities I am in no way advocating a return to the state of consciousness of this ancient Man. I do not believe it is "better", so to speak for we were that ancient man and the consciousness during those times is called atavistic in which his inner perception of numbers was a given in the same way as you and I can see a blue sky.

The consequence of this inner perception or atavism with respect to us is a lack of individual identity for the singular man. In essence his thought world or thinking was a "gift of the gods" as was the intellect as noted by the Greek philosophers.

Enter the age of abstractions and each individual man feels divorced from his surroundings and manifests a strong individuality. We now have an identity, an individuality, which in effect displays our freeing ourselves from that world to which the Greeks and ancients peoples could "see" naturally.

In essence "we have lost perception of the Divine", the ancient Egyptian would say. We have progressed beyond those "milk and honey days" and now stand tall as individual men. Concomitant with this is that everyone has a powerful attachment to his thoughts to which the truth arrives. We are all strong in intellect and to some reason, according to our inner comprehensions.

The abstractive abilities are secondary to the evolution of man, not Man secondary to this abstractive world, so to speak. But we are scientists in the best sense of the word and it is given to us the ability to break through the cosmic realities with our present intellectual and reasoning capabilities. We are a movement into the future and will not return to the atavistic state for this would be a backward and lost Man.

This activity has to be done consciously as we bring the intellect to a higher state of being: we are not fixed in time or space for we are cool.

I think I alluded to it in my initial post; in the beginning o, it then splits into oo to which the two entities felt like they can fly apart and then to the circular "three", or closed "three" I referenced on my beginning post.

http://wn.rsarchive.org/Lectures/GA2.../204-08-02.gif

Another look is that the "three" contains the "two" which contains the "one", with the 'one" primary which morphs into the number,qualitatively.

If we add or multiply abstractly it is easy to take the series out to "infinity" but if dealing with the "real" so to speak the mathematical procedures change.

This may give some insight for the shortest distance from point A to point B within Euclid is a straight line but in that world of the numbered reality there are no straight lines and the longest distance would be a straight line, if it could exist at all. I mentioned in the Platonic reference that Plato said Euclid's straight line or geometrical form is a abstracted presentation of its Platonic reality.

My initial impression is that there are no straight lines in the Platonic realm or realm of form and the triangle of 180 degrees would have to be curved.