Everytime a mathematician uses the concept of function or relation he is dealing with the concept of "being 2": he's relating one object to another. Everytime we use a logical conjuction we are indirectly making use of this basic concept so we cannot derive it from "being 1 and 1". Now, it seems possible to me to derive every other cardinality from these. "Being three" would mean "being 2 and being 1"; "being four" would mean "being three and being 1" and so on.


This question is quite vague (esoteric?), so the answer will not be very precise, too. First, I agree that "oneness" and "twoness" are fundamental because the former gives us the basic unit, the latter binary opposition and connection.

As you noticed, a relation r: X → Y is about "twoness". After all, it is formally defined as a subset of the Cartesian product X × Y = {(x, y): x ∈ X, y ∈ Y}.

A lot more complicated structures can be built from that, i.e. an arithmetic operation "⊙" on a set X (which may make it a group if certain requirements are met) could be defined by a function

g: M → X with M = X × X

and then setting

x₁ ⊙ x₂ := g(x₁, x₂).

At no point we need to bother with relating more than two sets or two elements with each other.

Now, seriously, we may have to sober up a bit... Perhaps this flight of thoughts got to our heads far too much?

If 1 and 2 are the basic concepts of mind, what difference is there between a child, that can count to two, and a great mathematician? That the great mathematician can juggle much better with 1s and 2s?

That seems preposterous.

And if we look at the natural numbers there is one important axiom, the axiom of mathematical induction:

If P is a proposition about natural numbers such that:

  • P is true for 0

  • for every natural number n, P being true for n implies that P is true for n + 1,

then P is true for every natural number.

we must remember that there are much simpler "toy" subsets of the usual axioms. They basically just define counting and calculating but can’t express general facts about numbers at all.

So how would we get to an idea like mathematical induction? Do we really grasp the infinite structure of the natural numbers itself from the concepts of 1 and 2?

No, it obviously doesn't work that way. 1 and 2 do not beget infinity.

So intuitively and as this simple example shows, there really is more to math, something we fundamentally cannot get from the concepts of 1 and 2.


There are many concepts in mathematics other than creating relations in functions, for instance geometry, vector theory etc. which require their own set of laws and seem to consist of their own basic concepts.

So, to answer your question bluntly NO; “being 1” and “being 2” are not basic concepts of our mind.

Furthermore, your propositions can't be used to explain all the numbers itself for instance 0 or negative numbers or irrational numbers and/or complex numbers which seem to appear equally basic to human mind, just like the oneness you mentioned.

  • On what basis do you regard complex numbers as equally basic to the human mind as oneness and twoness? Jan 6 '17 at 0:37
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    On what basis do we regard any number basic to human mind? The thing is all the numbers including real numbers such as one, two, three are equally abstract as any other complex numbers or even super geometric numbers. But the thing is right from the childhood we are taught the number systems and our mind has become comfortable to it. We think that these numbers really exist beyond our mind where as these are merely concepts which greatly help us understand process the information around us just like complex numbers does in other paradigms. Jan 6 '17 at 1:47
  • I reject the existence of abstract mathematical objects, so I'd begin by pushing back there. Descartes explains in the Meditations how mathematical concepts can be derived from thought. But I need not get psychologistic; a mathematical fictionalist is what I am. The point is that imaginary numbers are not more basic to the mind than oneness and twoness. Having to come up with something to generate a value for the square root of all Reals, as imaginary and complex numbers have to do, is probably an indicator of how non-basic of a concept it is. I mean, it's even fictive. Jan 6 '17 at 1:50
  • agreed. And there ways would reflect (or moreso 'just' reflect, or be closer to the core of reflecting) more basic structures than would complex/imaginary numbers. Jan 6 '17 at 1:51
  • @Lothrop Stoddard Sorry for deleting the comment it was an accident. It depends on what your definition of basic is. Real numbers are actually a subset of the complex numbers themselves. One could very well argue that complex numbers are the more basic concepts and we get real numbers when we square them. It just so happened that we grasped the concept of real numbers first. Later we realized that complex numbers are more fundamental and real numbers just happen to be a "special" case of the complex plane after we manipulate them in a special way instead of the other way round like you did. Jan 6 '17 at 1:59

The answer to your question is no, since malformed and ambiguous phrases cannot be unambiguous concepts, let alone those of our mind.

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