Number and Reality 3: The Demonstration That 2 Plus 2 Are 4 Spelled Out

(You will find below the demonstration, promised in the title of this post, that 2 plus 2 are 4. First, however, I have to say something about the post I have been promising, but failing, to publish for some time now, one setting out my understanding of realism and anti-realism in, on the one hand, the theory of universals and, on the other, the theory of numbers: I have not been able to work out to my satisfaction the understanding of the distinct philosophical motivations for the Platonist and Aristotelian realisms in the theory of universals which I thought I was just at the point of being able to work out. I am, therefore, ceasing here and now to promise that the post in question will be the next one to appear or even that it will appear at all. The lesson that I hope that I have learned: one should not commit the rookie error of promising a post until one has actually all but written the post.)

In the present post I will offer some (additional) preliminary remarks before going on to present for your review a demonstration, yes, a demonstration, indeed, a proof, that, given one evident assumption, 2 plus 2 are 4. Then, after having presented you with the proof, I will offer some postliminary comments.

Some Preliminaries. I’ll start with some remarks about the two premises of the proof. The first is that they, like all of the proof’s statements, are written in the style of symbolic or mathematical logic, using abbreviations like “&” for “and,” “~” for “it is not the case that,” “(x)” for “for any existent x,” and “Ixy,” for “x is identical with y.” If they had been written in anything like a standard English, without such abbreviations, the proof would not be anything any normal mortal would want to read. That being said, as written in something approaching a standard English the two premises read:

1. For any existent x and any existent y, x and y are two existents if and only if it is not the case that x is identical to y.

2. For any existent x, any existent y, any existent z, and any existent w, x, y, z, and w are four existents if and only if it is not the case that x is identical to y, it is not the case that x is identical to z, it is not the case that x is identical to w, it is not the case that y is identical to z, it is not the case that y is identical to w, and it is not the case that z is identical to w.

Next, it should be noted that the two premises are but definitions, the one defining what it is to be two existents and the other what it is to be four existents.

Next, about the assumption: the existents, a, b, c, and d that “instantiate” the variables x, y, z, and w, are none of them identical with any other of them; if, say, a were identical with b, then we would be dealing with a maximum of three existents.

Next, it should also be noted that, in addition to the two premises, the proof requires the resources of elementary logic alone to move to the conclusion.

Finally, a comment about what I had to do with the symbolization. As I have noted in a previous post, not knowing how to express, using the means available to users of this blog’s platform, WordPress, the “if and only if” relation in the double arrow notation standard in mathematical logic, I have to make do with with “iff.”

Enjoy!

The Proof

1. (x)(y)(2xy iff ~Ixy) [Pr., Def.]
2. (x)(y)(z)(w)(4xyzw iff (~Ixy & ~Ixz & ~Ixw & ~Iyz & ~Iyw & ~Izw)) [Pr., Def.]
3. 2ab iff ~Iab [1, U.I.]
4. 2cd iff ~Icd [1, U.I.]
5. (2ab –> ~Iab) & (~Iab –> 2ab) [3, P.E.]
6. (2cd –> ~Icd) & (~Icd –> 2cd) [4, P.E.]
7. 4abcd iff (~Iab & ~Iac & ~Iad & ~Ibc & ~Ibd & ~Icd) [2, U.I.]
8. (4abcd –> (~Iab & ~Iac & ~Iad & ~Ibc & ~Ibd & ~Icd)) [7, P.E.]
&
((~Iab & ~Iac & ~Iad & ~Ibc & ~Ibd & ~Icd) –> 4abcd)

9. ~Iab & ~Iac & ~Iad & ~Ibc & ~Ibd & ~Ibd [Ass., C.P.]
10. ((~Iab & ~Iac & ~Iad & ~Ibc & ~Ibd & ~Icd) –> 4abcd) [8, Comm.]
&
(4abcd –> (~Iab & ~Iac & ~Iad & ~Ibc & ~Ibd & ~Icd))

11. (2ab & 2cd) & (~Iac & ~Iad & ~Ibc & ~Ibd) [Ass., C.P.]
12. 2ab & 2cd [11, Simp.]
13. ~Iac & ~Iad & ~Ibc & ~Ibd [11, Comm., Simp.]
14. 2ab [12, Simp.]
15. 2cd [12, Comm., Simp.]
16. 2ab –> ~Iab [5, Simp.]
17. 2cd –> ~Icd [6, Simp.]
18. ~Iab [16, 14, M.P.]
19. ~Icd [17, 15, M.P.]
20. ~Iac & ~Iad [13, Simp.]
21. ~Ibc & ~Ibd [13, Comm., Simp.]
22. ~Iab & ~Iac & ~Iad [18, 20, Conj.]
23. ~Ibc & ~Ibd & ~Icd [21, 19, Conj.]
24. ~Iab & ~Iac & ~Iad & ~Ibc & ~Ibd & ~Icd [22, 23, Conj.]
25. (~Iab & ~Iac & ~Iad & ~Ibc & ~Ibd & ~Icd) –> 4abcd [10, Simp.]
26. 4abcd [25, 24, M.P.]

27. ((2ab & 2cd) & (~Iac & ~Iad & ~Ibc & ~Ibd)) –> 4abcd [11-26, C.P.]

28. 4abcd [Ass., C.P.]
29. 4abcd –> (~Iab & ~Iac & ~Iad & ~Ibc & ~Ibd & ~Icd) [10, Comm., Simp.]
30. ~Iab & ~Iac & ~Iad & ~Ibc & ~Ibd & ~Icd [29, 28, M.P.]
31. ~Iab –> 2ab [5, Comm., Simp.]
32. ~Icd –> 2cd [6, Comm., Simp.]
33. ~Iab [30, Simp.]
34. ~Icd [30, Comm., Simp.]
35. 2ab [31, 33, M.P.]
36. 2cd [32, 34, M.P.]
37. 2ab & 2cd [35, 36, Conj.]
38. ~Iab & ~Iac & ~Iad [30, Simp.]
39. ~Iac & ~Iad & ~Iab [38, Comm.]
40. ~Iac & ~Iad [39, Simp.]
41. ~Ibc & ~Ibd & ~Icd [30, Comm., Simp.]
42. ~Ibc & ~Ibd [41, Simp.]
43. ~Iac & ~Iad & ~Ibc & ~Ibd [40, 42, Conj.]
44. (2ab & 2cd) & (~Iac & ~Iad & ~Ibc & ~Ibd) [37, 43, Conj.]

45. 4abcd –> ((2ab & 2cd) & (~Iac & ~Iad & ~Ibc & ~Ibd)) [28-44, C.P.]

46. (((2ab & 2cd) & (~Iac & ~Iad & ~Ibc & ~I’d)) –> 4abcd) [27, 45, Conj.]
&
(4abcd –> ((2ab & 2cd) & (~Iac & ~Iad & ~Ibc & ~Ibd)))
47. ((2ab & 2cd) & (~Iac & ~Iad & ~Ibc & ~Ibd)) iff 4abcd [46, P.E.]

48. (~Iab & ~Iac & ~Iad & ~Ibc & ~Ibd & ~Ibd) [9-47, C.P.]
–>
(((2ab & 2cd) & (~Iac & ~Iad & ~Ibc & ~Ibd)) iff 4abcd)
49. (x)(y)(z)(w)((~Ixy & ~Ixz & ~Ixw & ~Iyz & ~Iyw & ~Iwz) [48, U.G.]
–>
(((2xy & 2zw) & (~Ixz & ~Ixw & ~Iyz & ~Iyw)) iff 4xyzw))

Q.E.D. (Writing out “Q.E.D,” was just plain fun.)

Some Postliminaries.

First, rendered into the almost standard English used above, the conclusion reads:

For any existents x, y, z, and w,
if x is not identical with y, x is not identical with z, x is not identical with w, y is not identical with z, y is not identical with w, and z is not identical with w,
then x and y are two existents and
it is not the case that x is identical with z and it is not the case that x is identical with w and it is not the case that y is identical with z and it is not the case that y is identical with w
if and only if
x, y, z, and w are four existents.

Second, 7 + 5 = 12, etc., can similarly be proven, but the proofs and their component propositions become rather imposing, though each of the steps needing to be taken remain quite elementary in nature.

Third, the proof is one belonging within the theory that I have dubbed ontological arithmetic, that part of the ontology of the identical as identical that comes with the inclusion of denials of identity. For example, the ontology of the identical as identical includes the principle of the symmetry of identity:

(x)(y)(Ixy –> Iyx)

or

For any existent x and any existent y, if x is identical with y, then y is identical with x.

It also includes:

(x)(y)(~Iyx –> ~Ixy)

or

For any existent x and any existent y, if y is not identical with x, then x is not identical with y.

Fourth, even as I prefer the term “ontology” over those of “metaphysics” and “first philosophy” which are part of the legacy of Aristotle’s Metaphysics, I still view the ontology of the identical as identical as, in principle, identical with the theory or science of the being as being of the Metaphysics. I devoted some thought to this identification in a couple of posts I published some years ago, the one the “A Note on Metaphysics, in Partial Response to Peter Hacker’s “Why Study Philosophy’” of January 27, 2014,

and the other the “Reading Alain Badiou’s Being and Event 5: Pluth on the Subject of Ontology” of October 21, 2013.

Fifth, to bolster my case that the theory or science of the being as being and the theory or science of the identical as identical are one and the same theory, I’ll add here the note that the affirmation that something is something is equivalent to the affirmation that that something is identical with that something. Thus, e.g.,

Donald Trump is the president of the United State.*

is equivalent to

Donald Trump is identical with the president of the United State.*

Fifth, the variables of ontological arithmetic, like those of the ontology of the identical as identical, range over real existents. Imagine, for example, that (1) I have two and only two coins in my left front pants pocket, which we can identify as coin a and coin b, coin a not being identical with coin b, and that I that I have two and only two coins in my right front pants pocket, which we can identify as coin c and coin d, coin c not being identical with coin d, (2) no coin is both a coin in my left front pants pocket and a coin in my right front pants pocket, and (3) I have no other coins, then it will be the case that I have exactly four coins. Thus it is that, in “Number and Reality 0. An Introduction to a Project in the Philosophy of Mathematics,” my post of October 24, 2020, I took delight in quoting the following line from Alfred North Whitehead’s An Introduction to Mathematics** (p. 9)

The nature of things is perfectly indifferent, of all things it is true that two and two make four.

There we go. Until next time.

Richard

*If only, as of the date of this writing, for seven or so more weeks.

**Alfred North Whitehead, An Introduction to Mathematics (Cambridge: Cambridge University Press, 1911)

About Rchard E. Hennessey

See above, "About the Author/Editor."
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