Philosophy Of Mathematics

[sigurdV]

First read about George Cantor

 

http://en.wikipedia.org/wiki/Georg_Cantor

 

Next check out what a cardinal number is:

 

http://en.wikipedia.org/wiki/Cardinal_number

 

 

There is a transfinite sequence of cardinal numbers:

 

The first transfinite number is [math]\aleph_0[/math] ,

 

Cantor extends arithmetics to transfinites:see Cardinal arithmetic.

 

Here is the rule for addition:

 

If X and Y are disjoint, addition is given by the union of X and Y.

 

Shouldnt "arithmetic" allow any "number" to add itself to itself?

That would mean that: [math]\aleph_0[/math] + [math]\aleph_0[/math] = [math]\aleph_0[/math]

 

Division is also defined, perhaps then [math]\aleph_0[/math] / [math]\aleph_0[/math] = 1

 

Dividing "[math]\aleph_0[/math] + [math]\aleph_0[/math] = [math]\aleph_0[/math] " by [math]\aleph_0[/math] we get 1+1 = 1

 

Any comments?

[sigurdV]

Shouldnt "Arithmetic" allow any "number" to add itself to itself?

The origin of the Mathemathical Concepts is hidden in prehistory: Perhaps already in the beginning of language there was a "one Many" system consisting in singular and plural,

but its not a system with operations...

 

That had to wait until "one" begot the successor "two"!

And to this day there are native tribes

using the the elementary "One Two Many" system.

Allowing a few arithmetic statements/truths...

 

Addition:

1 1+1=2

2 1+2=Many

3 2+2=Many

4 2+Many=Many

5 Many+Many=Many

 

Mathemathics begins with the profound statement:

 

1+1=2

Edited February 19, 2012 by sigurdV

[sigurdV]

Naive Mathemathics?

 

1+1=1

The Mighty Warrior asks the tribe Mathematician:

 

See here this drop of water in my left hand,

and the other drop in my right?

Now I add them together in one hand,

Do you call this drop TWO drops of water?

 

We must consider what we are adding,the Mathematician says.

(Thinking: The Club is yet mightier than the Thought.)

 

Drops of water arent behaving like apples,

but you might notice the drop is heavier

now when theres two drops in it.

 

Thinking:Maybe the warrior has a point here, maybe some objects cant be added to themselves,say: everything + everything...The result surely cant be more than everything?

Then there is the case of: nothing+nothing...

 

I guess they are not numbers...Hey! We HAVE only two numbers: the beginning number "one"

and the successor number "two"... "Many" is no number so it should be replaced...

Ive got it: its onetwo (=3) and then must come twotwo (=4) and next is onetwotwo! (=5)

Hmmm... Ive got onetwotwo fingers on my hand, and ive got two hands which means...Multiplication! :)

 

And the second number system was spread all over the world except to very far away places where the first wave colonisers still live this day...

[sigurdV]

Hi craig :) I love constructive critique! Notice the [math]\aleph_0[/math]

To be edited.(Its more like a letter to you than an ordinary post.)

 

I am moving towards Peano, but also i hope to go through logic and philosophy on the road.

(No maths without proofs,no proofs without logic and...) I have no clear route staked out on my map yet, so I dunno what will become of this...

 

Except that i want some matters belonging to Ontology cleared up :What Exists? What is Real?

For example: Statements,sentences... We can count sentenses/statements:

 

1 This is a sentence.

2 "This is a sentence" is a sentence

 

The first sentence added to the second sentence makes two sentences? Or?

 

3 "This is a sentence" = ""This is a sentence"is a sentence

 

So adding the sentences/statements makes no new sentence... Here again 1+1=1 Or?

 

PS How to get Music Notation?

 

PPS Perhaps not an obvious view, so i hasten to add that I think the statement

"There is a number" belongs first to ontology then secondly to mathematics.

[CraigD]

The first transfinite number is aleph null, i cant write it so i use

It’s a shame you can’t yet display pretty math notation to go along with your math thoughts, Sigurd, so it’s high time you learned to use LaTeX and math and imath tags. You could then write [math]\aleph_0[/math] with the greatest of ease, like I just did. :)

 

A quick way to get introduced to this feature is clicking Reply on some posts that have math that looks something like what you want, and seeing how whoever wrote it did the LaTeX. For reference and a deeper initiation, rhere are plenty of documents on the subject – my favorite are Wikipedia’s help:furmula page and John Forkosh’s L a T e X M a t h T u t o r i a l, which has a handy try-and-click box that speeds the learning process.

 

I use wikipedia a lot, and often grab formulae from it’s pages. Wikipedia keeps the LaTeX for its formulae in the ALT attributes of image file. You can see this stuff a several ways – an easy one with a recent Firefox browser is right-clicking the formula image and selecting View image info. The ALT attribute is labeled “associated text:” in the window that pops up, and can be selected and copied.

 

Any comments?

I surely will, being an old fan, though far from a well informed one, on transfinite numbers. It’ll take me some time to compose my thoughts into something post-worthy, though, and the workday is upon me, so I’ll be a while.

 

Mathemathics begins with the profound statement:

 

1+1=2

A fairly accurate statement, IMHO. :thumbs_up

 

It’s a bit more accurate, conventionally, to say mathematics starts with the number 1 exists, then go on to 1+1=2. This is an slightly archaic adapted excerpt from the Peano postulates, which most mathic folk agree, at least in some sense, is the “root of all math” – especially those who had an “intro to number theory” class in school (I’ve never seen or heard of one that didn’t get to Peano on or not long after day 1).

 

Peano arithmatic isn’t transfinite, but it underlies the mathematics of that is, and is worth – nay, essential – knowing.

[sigurdV]

Why Not take a look at Modern Mathematics?

 

http://en.wikipedia.org/wiki/Unary_numeral_system

http://en.wikipedia.org/wiki/Binary_numeral_system

http://en.wikipedia.org/wiki/Arithmetic#Decimal_arithmetic

http://en.wikipedia.org/wiki/Peano_axioms

 

Happily the Tribal Matematician scrutinizes his system:

 

1. To be a number was to be 1, or its successor number 2

 

I have now invented a method for creating successor numbers for any number! SO:

 

2. IF a frame,f, contains the number 1 and the successor to any number in f,

 

THEN f contains ALL numbers!

 

The Warrior has been listening in and says:

 

How does a "frame" fit into your

"picture"? Is f contained in f?

 

The Matematician wakes up and says:

Frames are not numbers,so f is not in f!

 

The Warrior concludes: Then f is contained in the frame

containing only the frames not containing themselves...Right?

 

http://en.wikipedia.org/wiki/Murphys_Law

 

The Mathematician (thinking really fast): Well Warrior...Frames are numbered, so there is no frame containing the frames not containing themselves. Instead there is a frame1 containing those frames and frame1 is not among them :)

 

 

http://en.wikipedia.org/wiki/Semantic_theory_of_truth

http://en.wikipedia.org/wiki/Object_language

http://en.wikipedia.org/wiki/Metalanguage

[Qfwfq]

This is an slightly archaic adapted excerpt from the Peano postulates, which most mathic folk agree, at least in some sense, is the “root of all math” – especially those who had an “intro to number theory” class in school (I’ve never seen or heard of one that didn’t get to Peano on or not long after day 1).

Actually, set theory is considered more fundamental and even a few other topics. Number theory can be based on these. The only thing that (recently, and due to the Russel paradox) has been placed more fundamental than sets is categories.

[sigurdV]

Actually, set theory is considered more fundamental and even a few other topics. Number theory can be based on these. The only thing that (recently, and due to the Russel paradox) has been placed more fundamental than sets is categories.

My intention is not to inquire as to the foundation of "Modern" Mathematics, it is rather to identify a natural order among the mathematical concepts as they occurred to early humans.

Old "counting systems" survive to this day among Native Tribes living in isolation, and they are easy to examine and order.

 

So why didnt I start with the first system: The "One,Many " as exemplified in the Singular and Plural forms of Language?

Simply because I wanted to start with the first system that separate Mathemathics

from Logic and Philosophy!

 

The first system introduces "number" and "set" in the "Principle of Selection":

 

1 Out of a many (=Set) can one (=number) object (=element) be selected.

 

This is roughly equivalent both to Peano axiom 1 and Cantors definition of a set.

But we have to wait for the successor system to get the statement "1+1=2"

containing Addition and Identity.

Addition certainly was useful to the Tribe, but analysing the Identity Relation was at the time probably seen as a silly effort.

 

Inventing multiplication in order to simplify the telling of big numbers seems natural,

so left out yet of basic arithmetics are only the Inverse Relations of Subtraction and Division.

 

Not introducing them already is a slight oversight,my excuse is that I wanted to introduce

Bertrand Russells Paradox and its Mathematical solution as soon as possible

 

And why use the archaic word "frame" for set? Well...because I believe Mathemathics to be the science of "form and number"! ("Form" being derived from "Frame".)

Edited February 19, 2012 by sigurdV