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Types of numbers - is there a fundamental type?

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ruveyn
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09 Sep 2012, 6:33 pm

AspieRogue wrote:
A vector is not a truly a "number", it is a collection of quantities which when combined have both a magnitude and direction. A Tensor however, is a function; which is not the same thing mathematically as a number.


The cleanest definition of a tensor is a multi-linear function that maps sever copies of a vector space along with several copies of its dual space into the reals or complex numbers. But in the older text books tensors were collections of components that transformed in a particular way going from one basis to another. So tensors were bundles of numbers.

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ruveyn
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09 Sep 2012, 6:36 pm

AspieRogue wrote:
A vector is not a truly a "number", it is a collection of quantities which when combined have both a magnitude and direction. A Tensor however, is a function; which is not the same thing mathematically as a number.


Vectors are somewhat like numbers: the can be added and subtracted and multiplied by numbers which effectively stretch or shrink them. A vector space is a ring over a number set.

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Robdemanc
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10 Sep 2012, 4:35 am

marshall wrote:
Robdemanc wrote:
TallyMan wrote:
I'm not sure exactly how to frame this question, but will have a go.

In the way one could say that positive integer numbers are a subset of all real numbers, positive or negative and fractional numbers; real numbers themselves could be described as being a subset of complex numbers because they consist of both a real and an imaginary part. Is there a type of number that embraces all other types of number? A fundamental type of number for want of a better way of expressing it?


I think you are asking for the highest level of abstraction in our number system, is that correct? Reading your question reminds me of object oriented programming. You are searching for the super class of all numbers?

Who knows. I would wonder if when finding that all ecompassing set of numbers you could then identify a larger set.....


The problem is in mathematics we have multiple-inheritance. The natural numbers can be thought of as a subclass of real numbers, or a subclass of ordinals, yet neither the class of ordinals nor the class of real numbers are subclasses of the other. The hierarchy branches in the upward as well as the downward direction. There is no strict upward chain of abstraction even though the development from natural numbers to complex numbers gives that impression.

The best ultra-abstract definition I can think of is that all full number systems require the development of peano aritmetic at some place in their logical construction. The induction principle is foundational to all number systems. In the ZFC system that most mathematicians refer to as their axiomatic foundation, the induction principle is included in the axiom of infinity.


Yeah its not as simple as inheritance in OOP, but it reminds me of it.



ruveyn
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10 Sep 2012, 5:30 am

marshall wrote:
AspieRogue wrote:
A vector is not a truly a "number", it is a collection of quantities which when combined have both a magnitude and direction. A Tensor however, is a function; which is not the same thing mathematically as a number.


But complex numbers are semantically 2 dimensional vectors. .


True. But they also have algebraic properties not possessed by vectors in general. One can add, subtract, multiply and divide complex number among themselves. They also satisfy the commutative, associative and distributive properties we associate with "genuine" numbers. Also there is an algebraic closed subset of the complex numbers which are isomorphic to the real number system. If one restricts the real and imaginary parts of complex numbers to integer values, we have the Gaussian Integers. and even Gaussian Primes. You can't do that with vectors in general.

Object Inheritance is not the best way of looking at number systems. One should pay attention to the axioms (or postulates) they satisfy. The best abstract way of looking at numbers is by way of category theory.

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marshall
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10 Sep 2012, 10:47 pm

ruveyn wrote:
marshall wrote:
AspieRogue wrote:
A vector is not a truly a "number", it is a collection of quantities which when combined have both a magnitude and direction. A Tensor however, is a function; which is not the same thing mathematically as a number.


But complex numbers are semantically 2 dimensional vectors. .


True. But they also have algebraic properties not possessed by vectors in general. One can add, subtract, multiply and divide complex number among themselves. They also satisfy the commutative, associative and distributive properties we associate with "genuine" numbers. Also there is an algebraic closed subset of the complex numbers which are isomorphic to the real number system. If one restricts the real and imaginary parts of complex numbers to integer values, we have the Gaussian Integers. and even Gaussian Primes. You can't do that with vectors in general.

Object Inheritance is not the best way of looking at number systems. One should pay attention to the axioms (or postulates) they satisfy. The best abstract way of looking at numbers is by way of category theory.

ruveyn


I agree.

I think all number systems should have the induction principle somewhere in their axioms. The positive integers are the most fundamental number system.



Ancalagon
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10 Sep 2012, 11:58 pm

TallyMan wrote:
So would this mean that in principle there is no ultimate fundamental type of number?

That depends on what you mean by fundamental. If you mean a kind of number that is basic and part of every other system of numbers in some way or another, it would be the natural numbers (1, 2, 3, ...), as marshall said. If you mean the limit of how far you can extend the natural numbers, then the complex numbers are it in some sense (they have 'algebraic closure' IIRC). It is possible to make a number system that has 4 dimensions like complex numbers have 2 (called the quaternions), but that system isn't commutative (a*b =/= b*a).

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Could future mathematicians define new and currently unthought of algebraic axioms?

Yep. If you're interested in more detail about this sort of thing, I'd recommend looking up abstract algebra (sometimes known as modern algebra). It isn't particularly concerned with numbers per se, and you get things like the algebraic structure of the rotational/reflective symmetries of a square.


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ruveyn
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11 Sep 2012, 11:13 am

marshall wrote:

I agree.

I think all number systems should have the induction principle somewhere in their axioms. The positive integers are the most fundamental number system.


The closest thing to an induction principle that fits all sizes is the well ordering theorem of any set. This is equivalent to the Axiom of Choice.

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marshall
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11 Sep 2012, 2:21 pm

ruveyn wrote:
marshall wrote:

I agree.

I think all number systems should have the induction principle somewhere in their axioms. The positive integers are the most fundamental number system.


The closest thing to an induction principle that fits all sizes is the well ordering theorem of any set. This is equivalent to the Axiom of Choice.

ruveyn


I don't see how AOC is necessary. Number systems need not be well-orderable, yet all allow for the construction of a well-orderable subset (without invoking AOC). I think the class of surreal numbers needs a stronger version of the AOC to be well-orderable because it has an inaccessible cardinal. Yet constructing the surreal numbers requires mathematical induction. The logical legitimacy of all number systems somehow hinges on the logical legitimacy of peano arithmetic.



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11 Sep 2012, 3:04 pm

I'm afraid the discussion has now gone beyond my level of mathematical education so I can't follow it. However, I've found the discussion stimulating and have enjoyed reading the posts. I will follow up on some of the posts and principles and do some internet based research. I have a better understanding of what I'm looking for now.


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ruveyn
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11 Sep 2012, 8:39 pm

marshall wrote:
ruveyn wrote:
marshall wrote:

I agree.

I think all number systems should have the induction principle somewhere in their axioms. The positive integers are the most fundamental number system.


The closest thing to an induction principle that fits all sizes is the well ordering theorem of any set. This is equivalent to the Axiom of Choice.

ruveyn


I don't see how AOC is necessary. Number systems need not be well-orderable, yet all allow for the construction of a well-orderable subset (without invoking AOC). I think the class of surreal numbers needs a stronger version of the AOC to be well-orderable because it has an inaccessible cardinal. Yet constructing the surreal numbers requires mathematical induction. The logical legitimacy of all number systems somehow hinges on the logical legitimacy of peano arithmetic.


AoC imples any set can be well ordered. Let such a well order be denoted by <. Here is a very general induction principle. Suppose P is true for the first element which must exist because < is a well ordering. If P(x) true for all x < y imples P(y). and this is true for all y, the P(y) is true for all y. Proof: Suppose there is a y such that P(y) is false. It isn't the first element of the set. So there is a least element y in the set for which P(y) is false and y is not the least. Consider the set of all x < y. P(x) must be true for those x because y is the least element for which P(y) is false. But by the induction principle P(x) true for all x < y implies P(y) is true. Contradiction. That is the proof.

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