161–170 of 211 posts
Odd, I was thinking about this earlier today. I came to the conclusion that an axiomatization of math sans numbers doesn't make sense because you have to have a certain number of axioms. I'm certain a counterargument could be made though.
I think numbers are wonderful and cool abstract thingies. We grow up being taught them (in my case Arabic numerals in base 10) and they become such part of our being that we start to think that we know what “5” is. Then I was introduced to Roman Numerals. Didn’t like those much. That one taught me that 4 is more related to 5 than it is 3. Then some of us learn a base 2 number system or base 16 number system and for a…
I would say I knew what 5 was even before I learned any notation or written language whatsoever.
The three main ideas are that a sortal
tells us what the essence of a thing is
tells us how to count things of that kind, which requires knowing which things are different and which are the same
tells us when something continues to exist, and when it goes out of existence
I think numbers are wonderful and cool abstract thingies. We grow up being taught them (in my case Arabic numerals in base 10) and they become such part of our being that we start to think that we know what “5” is. Then I was introduced to Roman Numerals. Didn’t like those much. That one taught me that 4 is more related to 5 than it is 3. Then some of us learn a base 2 number system or base 16 number system and for a…
There is no Antimemetics Division
I think numbers are wonderful and cool abstract thingies. We grow up being taught them (in my case Arabic numerals in base 10) and they become such part of our being that we start to think that we know what “5” is. Then I was introduced to Roman Numerals. Didn’t like those much. That one taught me that 4 is more related to 5 than it is 3. Then some of us learn a base 2 number system or base 16 number system and for a…
> What is five-ness really? It is everything that has some relation to something that has that same relation to something that has that same relation to something that has that same relation to something that has that same relation to some unique thing that does NOT have that same relation to anything.
Stephen Wolfram is so self-centred, I sometimes wonder if it's actually a stage character invented by a really talented comedian.
I guess one of the most fundamental difference between Wolfram’s model for fundamental physics and traditional physics is that Wolfram’s doesn’t have the concept of measure or of continuum at the fundamental level. Space and time, according to Wolfram’s model of the universe, are emerging properties of ‘the network’. Without such things as space and measures, there is no numbers in the fundamental “equations” that dr…
It's as if Stephen Wolframs mental horizon starts getting more and more restricted over the years since he tries to frame everything he sees in terms of his physical theory.
Arguably, number came from money. Not just the counting, but the generalization/fungibility - money can be exchanged for anything (unlike barter); number can represent a quantity of anything. Trade and money have more direct survival advantages than number, an evolutionary gradient for improving the cognitive capacity supporting this generalization
Shouldn't language predate money? Language or cave paintings or whatever, use discrete symbols to describe physical reality
Sidenote: the oldest writing found (so far) is of accounting...
I think numbers are wonderful and cool abstract thingies. We grow up being taught them (in my case Arabic numerals in base 10) and they become such part of our being that we start to think that we know what “5” is. Then I was introduced to Roman Numerals. Didn’t like those much. That one taught me that 4 is more related to 5 than it is 3. Then some of us learn a base 2 number system or base 16 number system and for a…
> Did we really know what “5” was then? What is five-ness really? Ah, but we do, and it’s very simple: it is what five cows an five fingers have in common in the most obvious sense. There’s no mystery there. Or, stated in another way, “fiveness is something you’ll be very sad not to see when you look at your hand.”
And with the Unit came the Successor, the primordial operation, for what is cleaved may ever be joined. From one comes two, from two comes three, and ever on until forever and always, with each successor given a name as a number.
And with these numbers came a set comprising them, and the set was Natural and good, and from it came many wondrous things.
For from repetition of the Successor came Addition, and the set was closed under Addition, and the Counter saw that this was good.
And from repetition of Addition came Multiplication, and the set was closed under Multiplication, and the Counter saw that this was good.
And from repetition of Multiplication came Exponentiation, and the set was closed under Exponentiation, and the Counter saw that this was good.
But if a thing can be done it can be undone. What is given can be taken away. If there is Addition there must be Subtraction. A shadow fell over the face of the Counter for under Subtraction the set was not closed.
Yet the set of Natural numbers had its closure under Subtraction, and this closure was another set named Integers, and the Counter saw that the Integers were good.
But if Addition of a Natural number has an inverse, so too must Multiplication by a Natural number, and this inverse was Division. A shadow passed again across the face of the Counter for under Division by a Natural number the set of Integers was not closed.
Yet the set of Integers had its closure under Division by a Natural number, and the closure was another set named Rational numbers, and the Counter saw that the Rational numbers were good and rejoiced at their scope, for between any two Rational numbers was an infinity of other Rational numbers, each with its own name.
But if Addition and Multiplication by Natural numbers have inverses, so too must Exponentiation, and indeed, so must the combination of Addition, Multiplication, and Exponentiation in a polynomial with Integer coefficients, and this inverse was the finding of Roots. A shadow passed again across the face of the Counter for almost never were the Roots of polynomials Rational.
Yet the Roots of polynomials with Integer coefficients gave rise to a new set, the set of Algebraic numbers, and the Counter saw that the Algebraic numbers were good and rejoiced at their scope, for the Algebraic numbers have complexities that delight and amaze, and each has its own name.
And yet.
Almost no number is Algebraic.
Almost every number belongs instead to a Transcendental realm where there are many terrors and almost nothing can be named.