Exploring the Origin of Numbers: Logic and Set Theory

[friend]

Where do numbers come from? What is the logical basis for the existence of numbers?

Are numbers defined in mathematical logic as the cardinality of set? For example, it would seem to me that 3 is defined as the cardinality of any set that has 3 elements.

IIRC it was Whitehead and Russel that tried to derive numbers and arithmatic in terms of set theory and logically proved that 1+1=2. How did they define numbers to begin with? I have no training in mathematical logic, and I would appreciate a little help. Thank you.

 

[phinds]

Where do numbers come from?

Our fingers?

 

[MathematicalPhysicist]

They can't prove a definition.

If '2' is defined as the succesor of 1 then 1+1=2.

Do I really need to remind you the old saying that goes like:
"God gave us the natural numbers all else is man made"

 

[Uvohtufo]

Hi, this is my first post on this website.

I don't think there is a logical basis for numbers. There have been attempts in history to base mathematics in logic, but I think those attempts have a kind of confused purpose. If mathematics has existed clearly for millenia, with clear methods of validity, logic can only describe mathematics, it cannot justify it.

Set theory is thought of by some to be a logical basis for mathematics. In set theory there are a few common ways to define numbers. One being something like this: Each number is just a set of the previous number, and the first number is a set containing the null set.

1 = {∅}

2 = {{∅}}

3 = {{{∅}}}

...

Another being each number is a set of every number before it.

1 = {∅}

2 = {{∅}, ∅}

3 = {{{∅} , ∅}, {∅}, ∅}

...

There is no end to the number of ways 'number' can be defined in set theory. The important trend to see is that the natural numbers have a kind of successor relationship, where for all x, there is another one that comes after.

 

[MathematicalPhysicist]

Hi, this is my first post on this website.

I don't think there is a logical basis for numbers. There have been attempts in history to base mathematics in logic, but I think those attempts have a kind of confused purpose. If mathematics has existed clearly for millenia, with clear methods of validity, logic can only describe mathematics, it cannot justify it.

Set theory is thought of by some to be a logical basis for mathematics. In set theory there are a few common ways to define numbers. One being something like this: Each number is just a set of the previous number, and the first number is a set containing the null set.

1 = {∅}

2 = {{∅}}

3 = {{{∅}}}

...

Another being each number is a set of every number before it.

1 = {∅}

2 = {{∅}, ∅}

3 = {{{∅} , ∅}, {∅}, ∅}

...

There is no end to the number of ways 'number' can be defined in set theory. The important trend to see is that the natural numbers have a kind of successor relationship, where for all x, there is another one that comes after.

To me these constructions just look as another symbolism for numbers like the Roman numerals system.

 

[Uvohtufo]

To me these constructions just look as another symbolism for numbers like the Roman numerals system.

Perhaps. Its interesting that you say that. Symbolism is a really interesting thing. By saying we have different symbols, it makes it sound like there is one thing that all of them are used to refer to. But, which thing in the world does ' 3 ' refer to? ' 3 ' doesn't really refer to anything in particular does it? We are the ones who use the symbol ' 3 '. What ' 3 ' represents, is more dependent on how we use the symbol than the objects designated ' 3 '.

Its misleading that 'number' is a noun at all. I think mathematics is a system of rules which we follow for practical reasons. Math is something we do. There is no 'what' a number is. Knowing what to do with a number is sufficient justification.

So, is a set theoretic definition of ' 3 ' just another symbol? I am not sure. Romans were using ' III ' in more or less the same way we use ' 3 '. The similar use is what gives the symbols the same reference. We don't use set theoretic definitions like we use numbers. By saying '3 := {{{∅}}}', I think we are only managing to demonstrate that that the obscure operations of set theory seem to correspond with the natural numbers.

 

[friend]

It seems to me that 3 is referring to the cardinality of any and all sets that have 3 elements. It does not matter how you construct the set, or which elements are used in the set, the cardinality will always be "3".

Given that, I wonder if the cardinality of a set can be related to the Dirac measure. The Dirac measure is 1 if a particular element belongs to a particular set. Then the cardinality of a set can be thought of as the addition of the Dirac measures for each element in the universe of discourse. You'll add a 1 for each element in the set and 0 for each element not in the set. Adding 1 for every element in the set gives the cardinality of the set. Has anyone every heard of that?

 

[micromass]

It seems to me that 3 is referring to the cardinality of any and all sets that have 3 elements. It does not matter how you construct the set, or which elements are used in the set, the cardinality will always be "3".

And how would you define whether the cardinality of a set is 3? Doesn't this require that you define 3 first?

Given that, I wonder if the cardinality of a set can be related to the Dirac measure. The Dirac measure is 1 if a particular element belongs to a particular set. Then the cardinality of a set can be thought of as the addition of the Dirac measures for each element in the universe of discourse. You'll add a 1 for each element in the set and 0 for each element not in the set. Adding 1 for every element in the set gives the cardinality of the set. Has anyone every heard of that?

Sure, that can be done. But I doubt it's very useful. I've never seen this used.

 

[friend]

And how would you define whether the cardinality of a set is 3? Doesn't this require that you define 3 first?

Well, let's see... numbers are only relevant to the algebra of arithmatic. So just as numbers form an ordered set, 0<1<2<3<4<..., perhaps sets can be arranged in an ordering such that some sets are "larger" than some but "smaller" than others. It doesn't mean anything to say there are 3 elements unless you mean that there are more than 2 and less than 4. Then, numbers can be just alternative symbols used to keep track of the property by which they are ordered.

Sure, that can be done. But I doubt it's very useful. I've never seen this used.

The Dirac measure is basically a way of indicating a set with a cardinality of 1. Then the Dirac measure would be a way of increasing the ordering of a set. That's kind of like a successor function, going from a set of one order to a set of a higher order by use of the Dirac measure as a successor function.

 

[MathematicalPhysicist]

Perhaps. Its interesting that you say that. Symbolism is a really interesting thing. By saying we have different symbols, it makes it sound like there is one thing that all of them are used to refer to. But, which thing in the world does ' 3 ' refer to? ' 3 ' doesn't really refer to anything in particular does it? We are the ones who use the symbol ' 3 '. What ' 3 ' represents, is more dependent on how we use the symbol than the objects designated ' 3 '.

Its misleading that 'number' is a noun at all. I think mathematics is a system of rules which we follow for practical reasons. Math is something we do. There is no 'what' a number is. Knowing what to do with a number is sufficient justification.

So, is a set theoretic definition of ' 3 ' just another symbol? I am not sure. Romans were using ' III ' in more or less the same way we use ' 3 '. The similar use is what gives the symbols the same reference. We don't use set theoretic definitions like we use numbers. By saying '3 := {{{∅}}}', I think we are only managing to demonstrate that that the obscure operations of set theory seem to correspond with the natural numbers.

Well as every abstract notion, we cannot fully grasp the idea behind a 'number', like 'love' or 'friendship'.
I mean sometimes the number indicates order sometimes it represents quantity in a set of elements.

 

[Uvohtufo]

Well as every abstract notion, we cannot fully grasp the idea behind a 'number', like 'love' or 'friendship'.
I mean sometimes the number indicates order sometimes it represents quantity in a set of elements.

I don't know. I disagree.

Words are tools not unlike physical tools, and it makes sense that a tool wouldn't have a single purpose (say counting, versus ordering). I think we do fully grasp what ever idea is behind the word (what else is there to be grasped aside from an idea?). If 'love' fails you, its more a limitation of the tool in a certain application than it is a limit of your comprehension.

 

[friend]

Well as every abstract notion, we cannot fully grasp the idea behind a 'number', like 'love' or 'friendship'.
I mean sometimes the number indicates order sometimes it represents quantity in a set of elements.

I can't imagine when quantity is not with respect to an ordering. One can grasp the idea of ordering without quantity, all we know is that it is more than some but less than others, without knowing the quantity. But it seems we cannot know the quantity of something without also knowing that it is more or less than other quantities. So I would take it that ordering takes precedence over quantity/numbering.

Is the cardinality of a set equivalent to quantity? Or are there circumstances where cardinality does not require the use of numbers? I'm thinking in terms of algebra that can consider whether x<y or y<x, without knowing the quantity of x or y. Are there similar considerations in set theory where we don't necessarily need to know the specific number value of the cardinality of sets but are still able to consider whether cardinality of one set is more or less than another set?

 

[MathematicalPhysicist]

Friend, you have Cantor's argument that shows that the power set of the natural numbers has cardinality greater than that of the natural numbers.

 

[chiro]

The one structure that doesn't have ordering by default is a set.

You can divide and integrate sets together however you want, but they don't have to have any kind of ordered comparison between them. You can add an extra structure (i.e. a relation) to add ordering, but it is by no means necessary.

 

[friend]

The one structure that doesn't have ordering by default is a set.

You can divide and integrate sets together however you want, but they don't have to have any kind of ordered comparison between them. You can add an extra structure (i.e. a relation) to add ordering, but it is by no means necessary.

Right, there's no ordering within a set, no necessary means of how to list the elements.

However, between different sets, there seems to be an obvious property of cardinality that distinguishes one set from another.

Like a number, the cardinality of a set is irrespective of the various kinds of elements that are listed in the set. As far as number/cardinality is concerned, you could list the elements as {1,2,3,...}. Then you can take the union and intersection of differing sets by denoting similar elements with the same number. Then there seems to be some correspondence between union and addition.

For example, the card[{1,2,3}$\cup${4,5}]=card{1,2,3}+card{4,5} = 3+2=5.

 

[pbuk]

You need to look before Whitehead and Russell to the work of Peano, and before him Dedekind who wrote a paper "Was sind und was sollen die Zahlen?" ("What are numbers and what should they be?"), a translation of which appears in his "Essays on the Theory of Numbers" available on Amazon.

Dedekind's insight and rigour is compelling, although the rigour does disappear at some crucial moments!

 

[pbuk]

Right, there's no ordering within a set, no necessary means of how to list the elements.

Indeed: formal expressions of arithmetical theory add a means to do this (the "successor function").

However, between different sets, there seems to be an obvious property of cardinality that distinguishes one set from another.

Yes again. Dedekind expresses this property somewhat poetically as what is left if you "entirely neglect the special character of the elements [of the set]; simply retaining their distiguishability and taking into account only the relations to one another in which they are placed by the [successor function]" and thus defines the natural numbers essentially as the set of cardinalities of all possible sets of "things". Interestingly he did not consider 0 to be a natural number, presumably because the empty set is not a set of "things".

 

[friend]

Indeed: formal expressions of arithmetical theory add a means to do this (the "successor function").

Yes again. Dedekind expresses this property somewhat poetically as what is left if you "entirely neglect the special character of the elements [of the set]; simply retaining their distiguishability and taking into account only the relations to one another in which they are placed by the [successor function]" and thus defines the natural numbers essentially as the set of cardinalities of all possible sets of "things". Interestingly he did not consider 0 to be a natural number, presumably because the empty set is not a set of "things".

If numbers are mapped to the cardinality of sets, and addition is mapped to the union of disjoint sets, then what set theoretic concept is the successor function mapped to?

 

[Devils]

You are in luck. In the 1800's, Russel & Whitehead used mathematical logic to prove 1+1=2. But it took several hundred pages.

These days its far easier to use ZFC. (Zermelo–Fraenkel set theory with the axiom of choice).
http://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory

And there's a theorem prover called Metamath the can do lots of math proofs using ZFC as the underlying axioms.
http://us.metamath.org/mpegif/mmset.html

The proof tree for "2+2=4" has a depth of 150 nodes.
http://us.metamath.org/mpegif/mmset.html#trivia

Many of the traditional math proofs are there, all from simple set theory. Some proofs have tens of thousands of logical inferences.

 

[yossell]

If numbers are mapped to the cardinality of sets, and addition is mapped to the union of disjoint sets, then what set theoretic concept is the successor function mapped to?

In set theory, cardinality of a set can be thought be thought of as an abstraction on sets: two sets have the same cardinality iff there's a 1-1, onto mapping between them. Such mappings could be logically defined without presupposing a concept of number. Now, the notion of a 1-1 onto mapping is formally an equivalence relation. Accordingly, it was once hoped that the cardinality of a set could be identified with the equivalence class of sets with the same cardinality. Unfortunately, early axiomatisations that allowed such sets (I'm looking at you, Frege) turned out to be inconsistent.

Since then, the set theoretic identifications of cardinalities with sets have contented themselves with making an arbitrary choice: a common choice has been to identify 0 with the empty set, 1 with the {{ }}, and, in general, n + 1 with n U {n}. Thus for the finite case, this is the set theoretic notion of successor. In general (allowing generalisation to the infinite case) cardinals are (certain) sets well ordered by the membership relation.

However, once we are dealing with infinite sets, this operation no longer generates a set with a greater cardinality. It is still used to generate sets representing the notion of an ordinal number, but that is another topic. Instead, the successor cardinal of an infinite cardinal A is identified with the set of all well orderings of A. It can be shown that such a set is the smallest cardinal whose cardinality is greater than A. Thus, amongst infinite cardinals, this set theoretic operation is the set theoretic version of successor.

 

[brianhurren]

call me drunk or mental or both, but I was told by a tutor in a computer class once that the only natural function that exists is the logical function '&' and since you can't have a nothing, there must be at least a '1' and since everything naturaly add to each other you must get '1 &' . if everything adds to everything else, and we accept that also includes adding to itself then we get "1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1..." all the natural numbers out to infinity. So natural numbers are really just a collection of ones that can be generated through the only natural logical function '&'

"Proof that not only is there a God but he is also drunk"

 

[HallsofIvy]

Well, I'll just say that computer class tutors are not the best source of information about fundamentals of mathematics! Although here, it is not too far off. Sounds like a rough description of Peano's axioms for the natural numbers.

 

[Uvohtufo]

call me drunk or mental or both, but I was told by a tutor in a computer class once that the only natural function that exists is the logical function '&'

Aside from 'natural function' being unclear...

In propositional logic, we typically have the symbols -, &, V, and → right? But interestingly enough you only need the negation and one of the other symbols, and from there you can derive the rest. I get the impression that a lot of computer or math people view & or V as the fundamental symbol, but philosophy and logic people often view → as the fundamental unit. This goes back to the origins of logic back in Greece where logic and reasoning were forms of arguments, and implication was an application of deductive reasoning.

Philosophers and logicians then investigated what the conditional (A → B) really conveys and built up symbolic logic from there. Eventually computer scientists and mathematicians adopting the same syntax.

 

[friend]

Philosophers and logicians then investigated what the conditional (A → B) really conveys and built up symbolic logic from there. Eventually computer scientists and mathematicians adopting the same syntax.

A really interesting connection between material implication, set theory, and numbers is the Dirac measure which returns the number 1 if a specific element is a member of a specific set. And if that element is not in that set, then the Dirac measure returns the number 0.

IIRC, implication can be represented by set inclusion. The existence of a set implies that its elements exist. For example, all men are mortal, and Socrates is a man, therefore, Socrates is mortal. The conclusion is implied because Socrates is included in the set of men who are mortal.

And the Dirac measure also seems to be a function that returns the cardinality - at least for a set with one element. So is there a way of equating the cardinality of larger sets in terms of Dirac measure which returns the cardinality of a single element set? I think this is the same as asking if there is an natural way to map set theoretic concepts to numerical arithematic operations?

 

[exponent137]

What about logic, which is used on classical or on a "theoretical" quantum computer? Can it be predecessor of natural numbers?