# What is a good approach to the question "the real number 2 is the same as the complex number 2?"

If Platonism holds, mathematical objects exist independently of our minds, and so the number two exists independently of our minds, but can there be multiple number 2's independent of our minds (e.g. one complex, and one real)?

Overall, is there a good place for some history/literature on this question? Has it been studied in detail?

• Real and complex numbers are not distinct; real numbers are a special case of complex numbers. So the 2s are the same not different. Apr 6 at 18:37
• Real numbers have properties that complex numbers do not have, see e.g. here, so you could argue that since the number systems are different, the respective structural contributors are ontologically distinct. See e.g. structuralism.
– apg
Apr 6 at 18:55
• but that only derives from being a special case; 2 is still 2. I think you are talking about the general case. Apr 6 at 19:01
• @WeatherVane, I don't think it's quite so trivial, but I do kind of agree when it comes to 2 - if we understand the different number kinds in terms of the axioms of the algebra that define them then the natural 2 is always the successor to 1, which is the multiplicative identity. But could we ask a very similar question about whether being the multiplicative identity means the same thing in the integer ring/ complex and real fields? Apr 6 at 19:23
• @Sandejo I should have said, R is a subset of C. It is a subset of C which is also a field. Apr 7 at 4:12

A Platonist with respect to what?

If you're a platonist with respect to sets, the real number 2 and the complex number 2 are in fact different. For the reals, one proceeds via, say, Dedekind cuts, for the complex numbers we realize the underlying set as R x R, then proceed to define addition and multiplication. In fact, 2 as a natural, integer, rational, real, complex, etc... are all in fact different- although there is a natural (set)-embedding from each structure to its superstructures.

This may seem strange, but its a consequence of Platonism + using, say, ZFC as a foundation.

Some (say Beneceraff) find this strange. You'll want to think of structuralism for a different perspective on the matter.

EDIT:

(a): by platonism, I mean something along the lines of existence, abstraction, and independence holding for "the" mathematical ontology. This may not be what the historical Plato thought, see Landry for further discussion. Nonetheless, this is typically what is meant in the literature today.

(b): Of course, once can also be a platonist about structures (ante rem).

(c): for an alternate platonistic view that does not priviledge any one set theoretic universe, one can also consider Balagauers plenitudinuous platonism. It consciously draws on model theoretic concepts, so that both the real and complex number two "satisfy" being the number two. Thus, there can be multiple number twos that exist independently of us.

• I disagree. You've shifted the argument from the special case to the general case. Of course complex numbers are not the same as real numbers, but the complex number 2 is the same in all respects as the real number 2. Apr 6 at 19:04
• But 2 + 2 + 0i = 4. In day to day actual mathematics, 2 = 2 + 0i. Apr 6 at 22:27
• @WeatherVane "the complex number 2 is the same in all respects as the real number 2." Patently false. As sets, they are distinct. I assume that's the point of the question. They're distinct sets but in some sense the same number. And the question is to nail down what "some sense" means here. Natural injection perhaps? That's the standard answer. Apr 7 at 3:36
• No. There is no single set of real numbers that is "the" real numbers, but any set with the appropriate cardinality can be endowed with the structure (algebraic and/or topological) of the real numbers, making it a copy of the real numbers. Merely because someone once defined the real numbers as a certain set doesn't mean that is the only set of real numbers there is. Apr 17 at 4:06

The thing is that for Plato the number two cannot be confused with the symbol that represents the number. This symbol that you print on a paper or even that you think is not the number two, but an instatiation of that number, it participates in the nature of the number two.

When a mathematician construct the real numbers (as Dedekind cuts or equivalent classes) with some set theory as foundation, he is representing the real matematics with a symbolic system that is not the mathematical object in itself. Formally speaking, one could say that the complex numbers are a different set from the reals and with use a inclusion map to make an isomorphic embeding, and this is true in the symbolic sense, but for a platonist both numbers, the real number two and the complex number 2 are representations or participations in the entity number two.

• The problem is that it is not just symbolic system of mathematicians that distinguishes 2 and 2+0i, they "participate" in different Platonic settings. That a monomorphism from one to the other can be defined by mathematicians no more proves that they are the same entity than that different constructions of real numbers can be given by mathematicians proves that there are multiple real numbers 2. Quadratic polynomials with real coefficients are even isomorphic to 3D space, that does not mean that a polynomial and a vector it is mapped to are the same Platonic entity either. Apr 8 at 18:48
• Yes, you have a good point. This depends of what one call by an "platonic entity". The form of Two cannot be reduced to the quantity two. When i have two symbols, even if they do not relate to the quantity two in its meaning, they have in the own writing the participation in the Two. Any duality in the world participates in the Two. The quantity two participates in the Two (but this does not mean that it exaust all the Two). Both, the real and the complex two have the participation in the Two by the aspect of quantity.
– LAU
Apr 10 at 10:53
• Second part. I'm not a platonist, but i think that one could represents the view of Plato whithout the need for the suposition of a entity for each mathematical construction. One can think that each construction reflects aspects of the real mathematical entities. Some platonists are more in the Pytagorian approach, some not.
– LAU
Apr 10 at 11:01