## 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)

## Number and Reality 2. Towards a Philosophy of Mathematics Bibliography

[This post is an exact replacement for the now deleted “Number and Reality 3.” One should not, I think we can all agree, have “Number and Reality 3” immediately succeed “Number and Reality 1” in a series devoted to number and reality.]

I am hard at work on the post or posts that I think will be the next one or two in the series of posts that I am devoting to the philosophy of mathematics. It or they will bear on the Platonist, realist, and nominalist theories in metaphysics, i.e., on their “solutions” to the so-called “problem of universals,” and on the philosophical theories immediately opposed to them. If all goes according to plan, it or they will be followed by a post or two bearing on the partially analogous Platonist, realist, and nominalist theories in the metaphysics or ontology of mathematics.

An announcement like the foregoing hardly justifies my having led you to read this far. In
recognition that you deserve something of rather more value for your doggedness, I will recommend that you take note of the post that Prof. Peter Smith’s has published on his blog, one that I have recently begun to follow, Logic Matters. Bearing the title, “Philosophy of mathematics — a reading list,” the post lists more than two dozen recent titles in the philosophy of mathematics and offers his expert’s summary evaluations, not all equally kind, of them.

Readers of this blog may recall that, in my post of October 24, 2020, “Number and Reality 0. An Introduction to a Project in the Philosophy of Mathematics,” I announced my intention to make use of Stewart Shapiro’s Thinking about Mathematics* as an “initial guide” as I re-explore and rethink my understanding of the philosophy of mathematics. I noticed, then, with particular interest what Smith has to say about Shapiro’s book.

Let’s begin [the list] with an entry-level book first published twenty years ago but not yet superseded or really improved on:
1. Stewart Shapiro, Thinking About Mathematics (OUP, 2000). After introductory chapters setting out some key problems and sketching some history, there is a group of chapters on what Shapiro calls ‘The Big Three’, meaning the three programmatic ideas that shaped so much philosophical thinking about mathematics for the first half of the twentieth century — i.e. varieties of logicism, formalism, and intuitionism. Then there follows a group of chapters on ‘The Contemporary Scene’, on varieties of realism, fictionalism, and structuralism. This might be said to be a rather conservative menu — but then I think this is just what is needed for a very first introduction to the area, and Shapiro writes with very admirable clarity.

He also had high praise for the major collection of essays** of which Shapiro was the editor, saying:

Stewart Shapiro (ed.) The Oxford Handbook of Philosophy of Mathematics and Logic (OUP, 2005). The editor’s introductory essay is in fact called ‘Philosophy of mathematics and its logic’, which should surely also have been the whole Handbook’s title — for of the twenty-six essays here, twenty are straight philosophy of mathematics, and the logic essays are mostly closely relevant to mathematics too. Large handbooks of this general type can often be very mixed bags, containing essays of decidedly varying quality; but this one really is a triumph. Some of the essays are very substantial, and as I recall it none is a makeweight. There are often pairs of essays taking divergent approaches (e.g. to contemporary logicism, to intuitionism, to structuralism). Of course, there are variations in the accessibility of the individual essays: but Shapiro seems to have done wonderful work in keepinAfg his very well-selected authors under control! So for any serious student now — perhaps beginning graduate student — this must be the place to start explorations of issues in more recent philosophy of mathematics.

Before closing, I’d like to draw your attention to another good deed that Smith has done for those engaged in, if not immediately the philosophy of mathematics, at least logic and thereby all branches of philosophy and indeed all branches of thought. That is, in an earlier post, “Free introductions to formal logic?” he laid out another list, “curated,” one of a higher degree of sophistication than yours truly might put it, of available introductory logic textbooks. The list includes some that are excellent, in his judgment, as well as free.

One of the logic texts, quite modestly entitled An Introduction to Formal Logic,*** is by Smith himself. He says about it:

Peter Smith, An Introduction to Formal Logic (2nd edition, originally CUP, 2020) Webpage here. Available also from Amazon print on demand. Doesn’t cover as much and more expansive than [the previously mentioned] forallx, so perhaps more accessible for self-study

There we go. Until next time.

Richard

*Stewart Shapiro, Thinking about Mathematics. The Philosophy of Mathematics (Oxford and New York: Oxford University Press, 2000). Thinking about Mathematics is readily available for purchase through Amazon.com. You need only click on the following image to be taken to the Amazon site:

Nota bene: As an Amazon Associate, I earn from qualifying purchases.

**Stewart Shapiro (ed.) The Oxford Handbook of Philosophy of Mathematics and Logic (Oxford and New York: Oxford University Press, 2005). The Oxford Handbook of Philosophy of Mathematics and Logic too is readily available for purchase through Amazon.com. Here too you need only click on the following image to be taken to the Amazon site:

Nota bene: As an Amazon Associate, I earn from qualifying purchases. You may also wish to note that I am not sure how long it will be before I will be making systematic use of this latter text in my posts.

***Peter Smith, An Introduction to Formal Logic (2nd edition, originally CUP, 2020).

Nota bene: As an Amazon Associate, I earn from qualifying purchases. You may also wish to note that I doubt that will be making systematic use of this text in my posts, excellent though it is.

## Number and Reality 1. Some Initial Questions and the Initial Question

1. In the immediately previous post, I observed that I now find myself enjoying the leisure that will allow me “to engage in some serious thinking in the philosophy of mathematics.” More specifically, I announced my intention “to begin the preparation necessary for the rewriting, the much needed rewriting, of Quantity and Reality. The Bases of an Aristotelian Philosophy of Mathematics,” the doctoral dissertation I wrote and defended nearly forty years age, but which, however, left me then and leaves me now quite dissatisfied. I also let it be known that, given that which I have begun to understand, the rewritten document would be so restricted as to warrant the title the present series of posts bears, Number and Reality, along with the sub-title, The Bases of an Aristotelian Philosophy of Arithmetic.

I further announced that I would be using Stewart Shapiro’s accessible and lucid introduction to the philosophy of mathematics, Thinking about Mathematics. The Philosophy of Mathematics,* as an initial guide for my reflections. And, as I closed the post, I invited any and all interested parties to join with me in those reflections.

It is my intention, then, using Shapiro’s Thinking about Mathematics as an initial guide, to begin the preparation necessary for the rewriting, the much needed rewriting, of Quantity and Reality and to record the progress that I hope to make in that preparation in the form of posts to this blog. And it is my hope that others will join with me in discussions of those reflections.

2. The aim specific to the present post is that of coming to an initial judgment about the logically first step that the philosophy of arithmetic motivating “Number and Reality” needs to take. I’ll have the coming to that initial judgment arise out of a reading of the first paragraph of the preface to Thinking about Mathematics. It reads:

This is a philosophy book about mathematics. There are, first of all, matters of metaphysics. What is mathematics all about? Does it have a subject matter? What is this subject matter? What are numbers, sets, points, functions, and so on? Then there are semantics matters. What do mathematical statements mean? What is the nature of mathematical truth? And epistemology. How is mathematics known? What is its methodology? Is observation involved, or is it a purely mental exercise? How are disputes among mathematicians adjudicated? What is a proof? Are proofs absolutely certain, immune from rational doubt? What is the logic of mathematics? Are there unknowable mathematical truths?

3. The opening two sentences raise several significant questions. The one is:

What is philosophy?

The second is:

What is mathematics?

The third is:

What is metaphysics?

A fourth is, or a few more questions are:

What are the relations between and among philosophy, mathematics, and metaphysics?

These are all questions that we will have to answer, at some future point in this series of posts. For now, however, let’s content ourselves with a possible answer to the question of what philosophy is, suggested by the rest of the paragraph, that philosophy is an intellectual activity that asks questions like the ones the paragraph poses.

3. The paragraph itself contains three sets of questions. The first set contains, as Shapiro tells us, questions about the metaphysics of mathematics, the second set contains questions about the semantics of mathematics, and the third set contains questions about the epistemology of mathematics (two more questions raise their head, about semantics and epistemology). In the present post we’ll do some initial thinking focused on the first set alone of the questions Shapiro himself asks, though that initial thinking will itself include yet more questions.

The first set of questions contains four questions. Now the first question, “What is mathematics all about?” is a complex question, complex in that it presupposes an affirmative answer to the logically prior question, “Is there something that mathematics is all about?” (I am, of course, not saying that this complex question is an example of the fallacious argument, or fallacy, of the complex question. Ditto for the further complex questions that we will take note of presently.)

The latter question, “Is there something that mathematics is all about?” is equivalent to Shapiro’s second question, “Does it have a subject matter?” This is so at least if that which mathematics is “all about” is identical with its “subject matter.” And, if that is the case, then the third question, “What is this subject matter?” is equivalent to the first, “What is mathematics all about?” It, the third question, is then also a complex question, one presupposing an affirmative answer to the second.

4. The fourth of Shapiro’s questions, “What are numbers, sets, points, functions, and so on?” gives us, if not a complete answer to the question of what the subject matter of mathematics is, at least a partial answer to the question of what it includes; it includes numbers, sets, points, functions, and so on. But we have also to take note that the question, quite evidently, is a compound question, the component questions of which are:

What are numbers?

What are sets?

What are points?

What are functions?

And so on.

I expect to be expected to offer answers to these and other questions as we proceed. But the questions just posed are also complex questions, presupposing affirmative answers to the questions:

Do numbers exist?

Do sets exist?

Do points exist?

Do functions exist?

And so on.

Thus, the affirmative answers:

Numbers exist.

Sets exist.

Points exist.

Functions exist.

And so on.

It is, I think, only if such entities are existents that they are capable of being whatever they are.

This would seem to be the place where we should notice that the title of Shapiro’s eighth chapter is, in fact, “Numbers exist” and that his preface tells us (pp. viii-ix):

Chapter 8 is about views that take mathematical language literally, at face value, and hold that the bulk of the assertions of mathematicians are true. These philosophers hold that numbers, functions, points, and so on exist independent of the mathematician. They then try to show how we can have knowledge about such items, and how mathematics, so interpreted, relates to the physical world.

5. I noted above that the question, “What are numbers?” and the similar questions about sets, points, functions, and so on are complex questions, presupposing affirmative answers to the questions, “Do numbers exist?” and the similar questions about sets, points, functions, and so on. Now, in the post immediately preceding this one, “Number and Reality 0. An Introduction to a Project in the Philosophy of Mathematics,” I also took note of the defining thesis of what I there nearly, though not quite, called realism in the ontology of mathematical objects. This is the thesis that Shapiro, in his Thinking about Mathematics* (p. 25), identified as “realism in ontology” and wanted us to define as “the view that at least some mathematical objects exist objectively, independent of the mathematician.” Let us, for present purposes, follow him, though using terminology that are in part mine and not fully his, and define realism in the ontology of numbers as “the view that at least some numbers exist objectively, independent of the mathematician.” Thus too, of course, for the realisms that are possible in the ontologies of sets, points, functions, and so on.

In, again, “Number and Reality 0. An Introduction to a Project in the Philosophy of Mathematics,” I also noted that I incline towards one of the two versions of realism that present themselves for our attention, that of an Aristotelian, or of a something like an Aristotelian realism, as opposed to a Platonist realism. About this difference there is more, much more, to come in future posts.

5. First, however, negative answers to those questions are also logically possible. And so let us define anti-realism in the ontology of numbers as the view that no numbers exist objectively, independent of the mathematician.” Thus too, of course, for the anti-realisms that are possible in the ontologies of sets, points, functions, and so on:

No numbers exist.

No sets exist.

No points exist.

No functions exist.

And so on.

This, of course, would seem to be the place where we should notice that the title of Shapiro’s ninth chapter counters that of the eighth chapter, announcing that, “No they don’t” and that his preface tells us (p. ix):

Chapter 9 concerns philosophers who deny the existence of specifically mathematical objects. The authors covered here either reinterpret mathematical assertions so that they come out true without presupposing the existence of mathematical objects, or else they delimit a serious role for mathematics other than asserting truths and denying falsehoods.

6. In the light of the above, I will close by presenting the provisional statement that:

There is a theory of numbers, arithmetic, and so of, first, whether any numbers exist and only then, if they do, of just what numbers are.

The aim specific to the present post, that of coming to an initial judgment about the logically first step that the series of posts, “Number and Reality,” needs to take, has been reached. It is that of answering the question of whether any numbers exist.

But wait. I described the statement given just above as provisional. And so it is. More needs to be said, some of which will be said in the next post.

There we go.

Until next time,

Richard

*Stewart Shapiro, Thinking about Mathematics. The Philosophy of Mathematics (Oxford and New York: Oxford University Press, 2000). Thinking about Mathematics is readily available for purchase through Amazon.com. You need only click on the following image to be taken to the Amazon site:

As an Amazon Associate I earn from qualifying purchases.

## Number and Reality 0. An Introduction to a Project in the Philosophy of Mathematics

1. At long last I find myself enjoying a set of circumstances that will allow me to engage in some serious thinking in the philosophy of mathematics; the two September posts, “Ontological Arithmetic. One Realistic Foot in the Door of the Philosophy of Mathematics,” and “Ontological Arithmetic. The Second Foot in the Door,” are the first early fruits, if “fruits” is the right word, of that thinking. Looking back further in time, the philosophy of mathematics was the area of my doctoral dissertation completed nearly four forty years ago; in the course of the intervening (long!) decades spent as a full-time academic administrator and an “off-and-on” adjunct professor of philosophy, I taught a wide variety of courses in philosophy (and others in Islamic Studies and a few in French), but none were immediately related to the philosophy of mathematics.

The reason that I am at this late date returning to the philosophy of mathematics is that even on the day I successfully, if that is what my being recognized as a doctor of philosophy in philosophy shows, defended my dissertation, I was deeply dissatisfied with it. And I am now even more dissatisfied with it. I will point to the reasons why below.

2. But first I’d like to note that the dissertation was not entirely without a few features that even now I think worth thinking about. One of them is evident in the dissertation’s title, Quantity and Reality, a title reflecting my admiration for the absolutely perfect title of Alfred North Whitehead’s Process and Reality, even though I was not, and am not, a “process philosopher.” More to point, however, it reflects the view that I had then and still have now, albeit perhaps with some precisions, that quantity is an attribute of real existents (or beings, things, or realities; in this context I use the four terms interchangeably, applying them to whatever is, in any way, whatsoever).

That is, first, I envisioned the theory I was seeking to articulate therein as a version of realism in the philosophy of mathematics, an initially apt enough definition of which has been set forth for us by Stewart Shapiro in his Thinking about Mathematics. The Philosophy of Mathematics* (p. 25).

At least on the surface, this theorem [i.e., “the ancient theorem that for every natural number n, there is a prime number m > n,” from which “[i]t follows that there is no largest prime number, and so there are infinitely many primes”] seems to concern numbers. What are these things? Are we to take the language of mathematics at face value and conclude that numbers, points, functions, and sets exist? If they do exist, are they independent of the mathematician, her mind, language, and so on? Define realism in ontology to be the view that at least some mathematical objects exist objectively, independent of the mathematician.

(I would have identified realism in the ontology of mathematics, rather than realism in ontology, as “the view that at least some mathematical objects exist objectively, independent of the mathematician,” for I distinguish between ontology and mathematics.)

One might say, for example, that the number two exists objectively, in independence from the mathematician.

3. But, second, realism in ontology tout court has traditionally, and with good reason, been divided into two sorts of realism, that of Platonism and that of Aristotelianism. Realism in the ontology of mathematics is, it has seemed to me, similarly to be divided. So, on the one hand, there is the realism of Platonism. Shapiro tells us (p. 27):

Realism in ontology does not, by itself, have any ramifications concerning the nature of the postulated mathematical objects (or properties or concepts), beyond the bare thesis that they exist objectively. What are numbers like? How do they relate to more mundane objects like stones and people? Among ontological realists, the more common view is that mathematical objects are acausal, eternal, indestructible, and not part of space-time. After a fashion, mathematical and scientific practice support this, once the existence of mathematical objects is conceded. The scientific literature contains no reference to the location of numbers or to their causal efficacy in natural phenomenon or to how one could go about creating or destroying a number. There is no mention of experiments to detect the presence of numbers or determine their mathematical properties. Such talk would be patently absurd. Realism in ontology is sometimes called ‘Platonism’, because Plato’s Forms are also acausal, eternal, indestructible, and not part of space-time.

To continue with the example, one might say that the number two exists, but nowhere in space and time.

Let us stipulate that Platonism is indeed and even by far the more common view of mathematical objects. I, however, was then and am now unable to accept Platonist realism either in ontology tout court or in the ontology of mathematics (in posts to come I will fulfill the obligation I have of explaining why). Moreover, I found it then and find it now necessary to accept an Aristotelian realism in ontology tout court, at least as I understood and understand what an Aristotelian realism should be (in a post to come I will fulfill the obligation I have here too of explaining why). I therefore thought, and think now, it necessary that an Aristotelian realism in the ontology of mathematics be worked out and presented; thus the subtitle of the dissertation, The Bases of an Aristotelian Philosophy of Mathematics.

4. The limitations of the dissertation are all too evident to me today. One thing that may be seen as a problem is the limitation of its scope. Though it did address at least in part a central problem, or confusion, inherent in the logicism of Gottlob Frege, Bertrand Russell, and Willard Van Orman Quine, in their identification of logic as the foundation of mathematics, the dissertation said very little about formalism and intuitionism, the other two of the doctrines that Shapiro identifies as “the big three” of the “major philosophical positions that dominated debates earlier” (p. vii) in the twentieth century (the “Contents” of Thinking about Mathematics shows the title of Part III as “THE BIG THREE.”) Nor did it say anything about the structuralism that is the perspective adopted by Shapiro himself; I was unaware of it.

But even within the scope of what I did take up there are real problems. One is that I was not able to work out the ontologicist, if I may, alternative to logicism. This is the theory that ontology, and not logic, provides the basic principles underlying arithmetic. Happily, I can report that I have been able recently to take a few initial steps in the working out of that ontologism; I offer in support of that claim that which I put forward in “Ontological Arithmetic. One Realistic Foot in the Door of the Philosophy of Mathematics” and “Ontological Arithmetic. The Second Foot in the Door.” I say, “a few initial steps’ advisedly; there are many more that need to be taken.

5. A second of the real problems falling within the scope of what I did take up in the dissertation has its basis in a difference that exists between, on the one hand, arithmetic and the theories, like algebra, that are the further developments of arithmetic and, on the other hand, geometry and its further developments. That is, on that one hand, some universal propositions of arithmetic, the true universal propositions of arithmetic, are exactly true, true without qualification, of any and all of the existents, beings, things, or realities denoted by their subjects, whether physical or not. Thus it is that, in his An Introduction to Mathematics** (p. 9), Alfred North Whitehead could say:

The first acquaintance which most people have of mathematics is through arithmetic. That two and two make four is usually taken as the type of a simple mathematical proposition which everyone will have heard of. Arithmetic, therefore, will be a good subject to consider in order to discover, if possible, the most obvious characteristic of the science. Now, the first noticeable fact about arithmetic is that it applies to everything, to tastes and to sounds, to apples and to angels, to the ideas of the mind and to the bones of the body. The nature of things is perfectly indifferent, of all things it is true that two and two make four. Thus we write down as the leading characteristics of mathematics that it deals with properties and ideas which are applicable to things just because they are things, and apart from any particular feelings. or emotions, or sensations, in any way connected with them. This is what is meant by calling mathematics an abstract science.

On that other hand, no universal propositions of geometry are exactly true, true without qualification, of any and all of the physical existents, beings, things, or realities purportedly denoted by their subjects. For but one example, not all physical lines are without thickness. Of course, one might reply that the lines that are properly the lines of geometry are ideal lines, and not physical. Then, however, one has the problem of determining what the relationship is between the ideal and the physical. Shapiro (pp. 69-70) sees here at least a potential problem.

There is a potential problem concerning the mismatch between real physical objects and geometric objects or properties. This, of course, is an instance of the mismatch between object and Form that motivates Platonism. Consider the brass sphere and the side of the ice cube. The sphere is bound to contain imperfections and the surface of the cube is certainly not completely flat. Recall theorem that a tangent to a circle intersects the circle in a single point…. This theorem is false concerning real circles and real straight lines. So what are we to make of Aristotle’s claim that ‘mathematical objects exist and are as they are said to be’, and the statement that ‘geometers speak correctly’?

This problem, as I will argue in a future post, is a problem that Aristotle and Aristotle never solved. It is also a problem that I have not yet solved.

6. To conclude this post: it is because I believe that I have, as the earlier posts mentioned above have indicated, gotten both of my feet in the door, albeit just barely, of the philosophy of arithmetic, while I have gotten neither foot at all in the door of the philosophy of geometry, that I have given the series of posts to which this one belongs the title, Number and Reality, as opposed to the Quantity and Reality of my dissertation. Correspondingly, I have been tempted to say that, if the series of post were to have a subtitle, that subtitle should be, “The Bases of an Aristotelian Philosophy of Arithmetic.” (It may be worth recognizing at the outset that, while I consider the philosophy of arithmetic I am in the process of working out to be an Aristotelian philosophy of arithmetic, the question of just how much it is in fact Aristotelian is one that perhaps many Aristotelians would consider to be at best, well, an open one.)

It is my intention, then, using Shapiro’s Thinking about Mathematics as an initial guide, to begin the preparation necessary for the rewriting, the much needed rewriting, of Quantity and Reality and to record the progress that I hope to make in that preparation in the form of posts to this blog. I will not take up all passages in Thinking about Mathematics or only passages of Thinking about Mathematics; for one thing, you may be sure that an adequate understanding of some of Shapiro’s passages will require, at least of me, some side journeys into a variety of blog posts, articles, and books.

But, of course, I hope for more than that. I hope that you will, not just follow, but also actively take part in the philosophical discussions among After Aristotle’s readers that the series of posts aims to have take place. I therefore invite you to ask for clarifications of, to challenge, to add to, or to otherwise comment on anything that I say in the posts to come.

There we go. Until next time,

Richard

*Stewart Shapiro, Thinking about Mathematics. The Philosophy of Mathematics (Oxford and New York: Oxford University Press, 2000). Thinking about Mathematics is readily available for purchase through Amazon.com. You need only click on the following image to be taken to the Amazon site:

As an Amazon Associate I earn from qualifying purchases.

**Alfred North Whitehead, An Introduction to Mathematics (Cambridge: Cambridge University Press, 1911). This book too is is readily available for purchase through Amazon.com. You need only click on the following image to be taken to the Amazon site:

You may wish to note that I do not anticipate returning to this book with any regularity.

## Logic Matters

The present post is a follow-up to my post of September 23, 2020, “Alfred Tarski on the Benefits of the Knowledge of Logic. A Timely Reminder.” In the post I presented a passage written by the logician, Alfred Tarki, spelling out why logic mattered circa 1940, that, as he put it, logic “makes men [sic] more critical—and thus makes less likely their being misled by all the pseudo-reasonings to which they are incessantly exposed in various parts of the world today.” It was my thought that, for the same reason, logic matters today.

This post was prompted by a post, “Free introductions to formal logic?” appearing in the blog, Logic Matters, published by Peter Smith, a retired professor of philosophy and logic at the University of Cambridge. Spoiler alert: The post answers its title’s question in the affirmative, with Smith taking note of five such books available, at no cost, for the downloading. The last book Smith lists is the second edition of his own An Introduction to Formal Logic (Second edition, Reprinted with corrections; Logic Matters, August 2020).

Logic, however, is useful only for those who know logic. So, I urge you to peruse Smith’s listing and those of the books listed you find most interesting.

Until next time.

Richard

## Ontological Arithmetic. The Second Foot in the Door

Greetings.

In the immediately previous post, the “Ontological Arithmetic. One Realistic Foot in the Door of the Philosophy of Mathematics” of September 24, 2020, I said that in the present post I would “spell out how one can prove, demonstrate, that one existent plus one existent are two existents, that two existents plus one existent are three existents, and that two existents plus two existents are four existents.” And that is what I will do, in four steps.

The Four Steps. The first step to be taken is that of proving that any existent is one existent.

I will slip three notes in here. The first is that such propositions as “any existent is one existent” stands in need of more precise reformulations, first in the “nearly standard English” and then in the abbreviations of “mathematical logic” that I made use of in the previous post. I’ll broaden the scope of that observation to include, in principle, all of the propositions playing a role in the coming proofs, although I will provide the reformulation in the “nearly standard English” of only a few of them, chiefly those serving as premises and conclusions.

The second note is that some may see in the proposition, that “any existent is one existent,” a statement of the classical scholastic metaphysical or ontological doctrine that “unity is a ‘transcendental’ property of being,” i.e., “all beings are units.”

The third note, more pertinent to the present post than the second note, is that the conclusion of the first step’s proof will be hereinafter identified as the Principle of Universal Unity, i.e., less grandly, the P.U.U., and will serve as a premise in the proofs that will constitute the second and third of the four steps.

The second step to be taken is that of proving that one plus one are two, while the third is that of proving that two plus one are three. The fourth step is that of spelling out how to go about proving that two plus two are four.

In the first three steps, then I do not simply spell out, how the three conclusions can be proven, I prove them. In the fourth step, I forego the full proof; it is rather lengthy. (Were I to offer a proof, directed to the attention of the scholars of the philosophy of mathematics of Immanuel Kant, that seven plus five are twelve….).

I. The Proof That Any Existent Is One Existent.

The Conclusion. First, I’ll state the conclusion of the proof here on offer, that any existent is one existent, in both its nearly standard English and its “mathematical logic” reformulations.

In the nearly standard English: For any existent x, x is one existent.

In the rendering of “mathematical logic”: (x)1x

The Premises. The proof has three premises.

The First Premise. The Principle of the Symmetry of Identity. The principle of the symmetry of identity (hereinafter, P.S.I.) reads:

In the nearly standard English: For any existent x and any existent y, if x is identical with y, then y is identical with x.

In the rendering of “mathematical logic”: (x)(y)(Ixy –> Iyx)

The Second Premise. The Principle of the Transitivity of Identity. The principle of the transitivity of identity (hereinafter, P.T.I.) reads:

In the nearly standard English: For any existent x, any existent y, and any existent z, if x is identical with y and y is identical with z, then x is identical with z.

In the rendering of “mathematical logic”: (x)(y)(z)((Ixy & Iyz) –> Iyx)

The Third Premise. The Definition of What It Is to Be One Existent.

In the nearly standard English: For any existent x, x is one existent if and only if it is not the case that there is an existent y and there is an existent z such that y is not identical with z and such that both y is identical with x and z is identical with x.

In the rendering of “mathematical logic”: (x)(1x iff ~(Ey)(Ez)(~Iyz & (Iyx & Izx)))

(As I I noted in the immediately 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.” And, while I’m at it, not knowing, again, how to express the backwards “E,” meaning “There exists…,” using the resources offered by tWordPress, I have had to make do with the standard “E.”)

It would help no one if I were to explicitly spell out every step in any of the four proofs using our nearly standard English. So, I will express the several steps of the first three proofs in the renderings of “mathematical logic.” All of the steps taken in any of the proofs are quite elementary and can be understood by anyone who has studied elementary “mathematical logic” or “symbolic logic.”

The Proof That Any Existent Is One Existent.

1. (x)(y)(Ixy –> Iyx) [Pr., P.S.I.]
2. (x)(y)(z)((Ixy & Iyz) –> Ixz) [Pr., P.T.I.]
3. (x)(1x iff ~(Ey)(Ez)((Iyx & Izx) & ~Iyz) [Pr., Def.]
4. 1a iff ~(Ey)(Ez)((Iya & Iza) & ~Iyz) [3, U.I.]
5. (1a –> ~(Ey)(Ez)((Iya & Iza) & ~Iyz)) & (~(Ey)(Ez)((Iya & Iza) & ~Iyz)) –> 1a) [4, L.E.]
6. (~(Ey)(Ez)(Iya & Iza) & ~Iyz)) –> 1a) [5, Comm.]
&
(1a –> ~(Ey)(Ez)((Iya & Iza) & ~Iyz)))
7. ~(Ey)(Ez)((Iya & Iza) & ~Iyz)) –> 1a [6, Simp.]

Now here, in Step 8, one assumes the contradictory of the anticipated conclusion and then derives a contradiction from the conjoining of that assumption to the premise set. That proves that that contradictory of the anticipated conclusion is inconsistent with the premise set and thus that the anticipated conclusion necessarily follows from the premise set.

8. ~1a [Ass., C.P.]
9. ~~(Ey)(Ez)((Iya & Iza) & ~Iyz)) [7, 8, M.T.]
10. (Ey)(Ez)((Iya & Iza) & ~Iyz)) [9, D.N.]
11. (Iba & Ica) & ~Ibc [10, E.I.]
12. Iba & Ica [11, Simp.]
13. Iba [12, Simp.]
14. Ica [13, Comm., Simp.]
15. Ica –> Iac [1, U.I.]
16. Iac [15, 14, M.P.] {Correction, 11/14/2020, “P” for “T”}
17. Iba & Iac [13, 16, Conj.]
18. (Iba & Iac) –> Ibc [2, U.I.]
19. Ibc [18, 17, M.P.]
20. ~Ibc [11, Comm., Simp.]
21. Ibc & ~Ibc [19, 20, Conj.]

22. ~1a –> (Ibc & ~Ibc) [8-21, C.P.]
23. ~(Ibc & ~Ibc) [Taut.]
24. ~~1a [22, 23, M.T.]
25. 1a [24, D.N.]
26. (x)1x [25, U.G.]

The Proof That One Existent Plus One Existent Are Two Existents.

1. (x)1x [Pr., P.U.U.]
2. (x)(y)(2xy iff ~Ixy) [Pr., Def.] {Correction, 11/14/2020, Insertion of “iff”}
3. 2ab iff ~Iab [2, U.I.]
4. (2ab –> ~Iab) & (~Iab –> 2ab) [3, L.E.]

“And,” “Plus,” and the Distinction Condition. I interrupt this proof to note that in Step 5, the first of two applications of the conditional proof technique, there are two propositions conjoined by the second “&” (i.e., of course, “and”). The first of the two conjuncts, the “(1a & 1b),” is itself a conjunction of two propositions, “1a” and “1b.” Were that the entire story, then the common way of stating the arithmetical truth, “One and one are two,” would be acceptable. It is not, however, acceptable, for one and one not need be two. If, for example, a and b are the same person, under different names, then one and one are one. And so there is the need for the second conjunct of the whole conjunction, “~Iab,” a recognition that a is not identical with b. It is because of this, as I have taken to call it, distinction condition, that I want to insist on the formulation, “one plus one are two,” rather than “one and one are two.”

5. (1a & 1b) & ~Iab [Ass., C.P.]
6. ~Iab –> 2ab [4, Comm., Simp.]
7. ~Iab [5, Comm., Simp.]
8. 2ab [6, 7, M.P.]

9. ((1a & 1b) & ~Iab) –> 2ab [5-8, C.P.]

10. 2ab [Ass., C.P.]
11. 2ab –> ~Iab [4, Simp.]
12. ~Iab [11, 10, M.P.]
13. 1a [1, U.I.]
14. 1b [1, U.I.]
15. 1a & 1b [13, 14, Conj.]
16. (1a & 1b) & ~Iab [15, 12, Conj.]

17. 2ab –> ((1a & 1b) & ~Iab) [10-16, C.P.]
18. (((1a & 1b) & ~Iab) –> 2ab) & (2ab –> ((1a & 1b) & ~Iab)) [9, 17, Conj.]
19. ((1a & 1b) & ~Iab) iff 2ab [18, L.E.]
20. (x)(y)(((1x & 1y) & ~Ixy) iff 2xy) [19, U.G.]

The Proof That Two Existents Plus One Existent Are Three Existents.

1. (x)1x [Pr., P.U.U.]
2. (x)(y)(2xy iff ~Ixy) [Pr., Def.]
3. (x)(y)(z)(3xyz iff (~Ixy & ~Ixz & ~Iyz)) [Pr., Def.]
4. 2ab iff ~Iab [2, U.I.]
5. (2ab –> ~Iab) & (~Iab –> 2ab) [4, L.E.]
6. 3abc iff (~Iab & ~Iac & ~Ibc) [3, U.I.]
7. (3abc –> (~Iab & ~Iac & ~Ibc)) & ((~Iab & ~Iac & ~Ibc) –> 3abc) [6, L.E.]

Let’s notice that in Step 8, there are two conjunctions of conjunctions. The second such conjunction of conjunctions, “~Iac & ~Ibc,” is the expression of the distinction condition for the addition at hand, that of two existents plus one existent. The distinction condition, one can surmise, is a function of the addenda.

8. (2ab & 1c) & (~Iac & ~Ibc) [Ass., C.P.]
9. 2ab & 1c [8, Simp.]
10. 2ab [9, Simp.]
11. 2ab –> ~Iab [5, Simp.]
12. ~Iab [11, 10, M.P.]
13. ~Iac & ~Ibc [8, Comm., Simp.]
14. ~Iac [13, Simp.]
15. ~Iab & ~Iac [12, 14, Conj.]
16. ~Ibc [13, Comm., Simp.]
17. ~Iab & ~Iac & ~Ibc [15, 16, Conj.]
18. (~Iab & ~Iac & ~Ibc) –> 3abc [7, Comm., Simp.]
19. 3abc [18, 17, M.P.]

20. ((2ab & 1c) & (~Iac & ~Ibc)) –> 3abc [8-19, C.P.]

21. 3abc [Ass., C.P.]
22. 3abc –> (~Iab & ~Iac & ~Ibc) [7, Simp.]
23. ~Iab & ~Iac & ~Ibc [22, 21, M.P.]
24. ~Iab –> 2ab [5, Comm., Simp.]
25. ~Iab [23, Simp.]
26. 2ab [24, 25, M.P.]
27. 1c [1, U.I.]
28. 2ab & 1c [26, 27, Conj.]
29. ~Iac & ~Ibc [23, Comm., Simp.]
30. (2ab & 1c) & (~Iac & ~Ibc) [28, 29, Conj.]

31. 3abc –> ((2ab & 1c) & (~Iac & ~Ibc)) [21-30, C.P.]
32. (((2ab & 1c) & (~Iac & ~Ibc)) –> 3abc) [20, 31, Conj.]
&
(3abc –> ((2ab & 1c) & (~Iac & ~Ibc)))
33. (((2ab & 1c) & (~Iac & ~Ibc)) iff 3abc [32, L.E.]
34. (x)(y)(z)(((2xy & 1z) & (~Ixz & ~Iyz)) iff 3xyz) [33, U.G.]

The Proof That 2 Plus 2 Are 4.

It would help no one if I were to explicitly spell out every step in the fourth proof even using the renderings of mathematical or symbolic logic. It will not help those who are not adept in that logic. And it will not help those who are adept in that logic, for they will not need the explicit spelling out, as each of the (very many) steps is utterly elementary. I’ll merely suggest to them that the use of conditional proofs, as was done in the previous proof, provides an easy way to the conclusion.

I’ll just get things started.

1. (x)(y)(2xy iff ~Ixy) [Pr., Def.]
2. (x)(y)(z)(z)(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. 4abcd iff (((~Iab & ~Iac & ~Iad) & (~Ibc & ~Ibd)) & ~Icd) [2, U.I.
6. (2ab iff ~Iab) & (~Iab  2ab) [3, L.E.
7. (2cd iff ~Icd) & (~Icd  2cd) [4, L.E.
8. (4abcd –> (((~Iab & ~Iac & ~Iad) & (~Ibc & ~Ibd)) & ~Icd)) [5, L.E.]
&
((((~Iab & ~Iac & ~Iad) & (~Ibc & ~Ibd)) & ~Icd) –> 4abcd)

Note the distinction condition operative in the second conjunction of the proof’s next step.

9. (2ab & 2cd) & (~Iac &~ Iad & ~Ibc & ~Ibd)) [Ass., C.P.]
.

.
.
45. (x)(y)(z)(w)((((2xy & 2zw) & (~Ixz & ~Ixw & ~Iyz & ~Iyw))) iff 4xyzw) [44, U.I.]

Summing Up.

In this post I have, as promised in the previous post, spelled out how one can prove, demonstrate, that one existent plus one existent are two existents, that two existents plus one existent are three existents, and that two existents plus two existents are four existents; in fact, I have provided complete proofs of the conclusions of the first two.

But I also stated, in the previous post, that in the present post I [would] “affirm that the philosophical realism of the ontology and arithmetic at hand is an apodicticism, according to which they are, not just theories of extra-mental and extra-linguistic existents, but demonstrative sciences thereof. With that done, I believe[d,] I [would] have two realistic feet in the door of the philosophy of mathematics.” My thinking now, however, is that I should do more than so affirm before allowing myself to believe that I have two realistic feet in the door of the philosophy of mathematics. So it is my intention to say, in the next post, or perhaps in the post after next, a little bit more about the philosophical understanding that the realistic philosophy of mathematics that I have in mind brings with it.

Until next time.

Richard

## Ontological Arithmetic. One Realistic Foot in the Door of the Philosophy of Mathematics

The aim of the present post is three-fold. I wish first to draw attention to the ontological theory of identity, i.e., of existents as identical with existents, and three quite basic principles of the theory. I then wish to then to draw attention to four arithmetical definitions that the introduction of non-identity into the ontology makes possible, statements, that is, of what it is to be one existent, what it is to be two existents, what it is to be three existents, and what it is to be four existents. There is little if anything in this post that is truly original, though the accompanying philosophical realism, according to which the ontology and arithmetic in question are theories of real existents, may startle some.

I am more inclined, though still only inclined, to believe that the results which will be presented in the next post are, though absolutely elementary, yet original; I say “inclined,” because I just don’t know if others have preceded me in arriving at them. That is, in the next post, I will spell out how one can prove, demonstrate, that one existent plus one existent are two existents, that two existents plus one existent are three existents, and that two existents plus two existents are four existents. It will follow that the philosophical realism of the ontology and arithmetic at hand is an apodicticism, according to which they are, not just theories, but demonstrative sciences. [Correction, December 10, 2020: The proof that two existents plus two existents are four existents was not published until my “Number and Reality 3: The Demonstration That 2 Plus 2 Are 4 Spelled Out,” posted on December 5, 2020.]

Ontology. There is, then, an ontology of identity. I will state the three basic principles of the theory in, first, a nearly standard English and then in a rendering typical of what you might find in a textbook of (so-called) “mathematical logic” (but I would rather identify as one of ontology); the nearly standard English rendering is for those who are not familiar with contemporary elementary logic and the rendering of “mathematical logic” is for those who are. I’ll also offer exemplifications of each of the three principles.

The Principle of the Reflexivity of Identity:

In the nearly standard English: For any existent x, x is identical with x.

In the rendering of “mathematical logic”: (x)(Ixx)

E.g.: Donald Trump is identical with Donald Trump.

The Principle of the Symmetry of Identity:

In the nearly standard English: For any existent x and any existent y, if x is identical with y, then y is identical with x.

In the rendering of “mathematical logic”: (x)(y)(Ixy —> Iyx)

E.g.: If Donald Trump is identical with the President of the United States, then the President of the United States is identical with Donald Trump.

The Principle of the Transitivity of Identity:

In the nearly standard English: For any existent x, any existent y, and any existent z, if x is identical with y and y is identical with z, then x is identical with z.

In the rendering of “mathematical logic”: (x)(y)(z)((Ixy & Iyz) —> Ixz)

E.g.: If Donald Trump is identical with the President of the United States and the President of the United States is identical with the Commander in Chief of the United States’ Army and Navy, etc., then Donald Trump is identical with the Commander in Chief of the United States’ Army and Navy, etc.

All extra-mental and extra-linguistic existents, including Donald Trump, are precisely that, extra-mental and extra-linguistic existents, and so we can note that the ontology of identity is a theory of extra-mental and extra-linguistic existents. To underline the point: the objects of the theory are not intra-mentally existent (existing in the mind) thoughts about the real or thoughts instead of the real; and they are not intra-linguistically existent (existing in language) words about the real or words instead of the real. We can also note, and not merely parenthetically, that the three principles just set forth are all absolutely true and true of absolutely everything.

(In the following expressing of the definitions in the “rendering of ‘mathematical logic’,” I found myself not knowing how to express, with the means available to WordPress users, the “if and only if” relation using the standard double arrow. So I had to make do with “iff.” I similarly found myself not knowing how to express the existential quantifier using the backwards capital letter “E” customary among logicians, and so had to make do with the capital letter “E” facing in the direction customary in the rest of the world.)

Ontological Arithmetic. Four Definitions. The three principles just placed before you are principles bearing upon identity. If we add non-identity to our ontology, such that one existent is not another, we will find ourselves having entered the realm of ontological arithmetic, via the following four arithmetical definitions.

The Definition of What It Is to Be One Existent.

In the nearly standard English: For any existent x, x is one existent if and only if it is not the case that there is an existent y and there is an existent z such that y is not identical with z and such that both y is identical with x and that z is identical with x.

In the rendering of “mathematical logic”: (x)(1x iff ~(Ey)(Ez)(~Iyz & (Iyx & Izx)))

[Note added, December 10, 2020: In the weeks that followed this post, I have grown less and less satisfied with this attempt at defining what it is to be one existent.]

The Definition of What It Is to Be Two Existents (A definition rather simpler than the previous).

In the nearly standard English: For any existent x and any existent y, x and y are two existents if and only if
x is not identical with y.

In the rendering of “mathematical logic”: (x)(y)(2xy iff ~Ixy)

The Definition of What It Is to Be Three Existents.

In the nearly standard English: For any existent x, any existent y and any existent z, x, y and z are three existents if and only if
x is not identical with y, x is not identical with z, and y is not identical with z.

In the rendering of “mathematical logic”: (x)(y)(z)(3xyz iff (~Ixy & ~Ixz & ~Iyz))

The Definition of What It Is to Be Four Existents.

In the nearly standard English: 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 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.

In the rendering of “mathematical logic”: (x)(y)(z)(w)(4xyzw iff (~Ixy & ~Ixz & ~Ixw & ~I yz & ~Iyw & ~Izw))

I could continue indefinitely, but I trust that that will not be necessary.

Summing Up. In the present post I have drawn attention to the ontological theory of identity, i.e., of existents as identical with existents, and to three quite basic principles of the theory. I have further drawn attention to four arithmetical definitions that the introduction of non-identity into the ontology makes possible, i.e., of what it is to be one existent, what it is to be two existents, what it is to be three existents, and what it is to be four existents.

With those definitions, I think it safe to say that I have one realistic foot in the door of the philosophy of mathematics.

In the next post, as I said above, I will spell out how one can demonstrate that one existent plus one existent are two existents, that two existents plus one existent are three existents, and that two existents plus two existents are four existents; I could continue indefinitely, but I trust that that will not be necessary. I will affirm that the philosophical realism of the ontology and arithmetic at hand is an apodicticism, according to which they are, not just theories of extra-mental and extra-linguistic existents, but demonstrative sciences thereof. With that done, I believe I will have two realistic feet in the door of the philosophy of mathematics.

Until next time.

Richard

## Alfred Tarski on the Benefits of the Knowledge of Logic. A Timely Reminder

The specifics of the problems facing the world and most occupying the mind of the great logician, Alfred Tarski, some 80 years ago, were certainly different from those now facing the United States of America and occupying our minds. Despite that, however, that which he wrote in 1940 in the preface to his classic Introduction to Logic and to the Methodology of Deductive Sciences still rings true.

I have no illusions that the development of logical thought, in particular, will have a very essential effect upon the process of the normalization of human relationships; but I do believe that the wider diffusion of the knowledge of logic may contribute to the acceleration of the process. For, on the one hand, by making the meaning of concepts precise and uniform in its own field and by stressing the necessity of such precision and uniformization in any other domain, logic leads to the possibility of better understanding among those who have the will for it.  And, on the other hand, by perfecting and sharpening the tools of thought, it makes men [sic] more critical—and thus makes less likely their being misled by all the pseudo-reasonings to which they are incessantly exposed in various parts of the world today.

Alfred Tarski, Introduction to Logic and to the Methodology of Deductive Sciences. (3rd edition; New York: Oxford University Press, 1965 [1941]), p. xv.

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## Reflecting on Russell’s Religion and Science 3. Russell as a Skeptic, Even with Respect to Science

1. The present post is the third in a series of posts reflecting on the philosophical theses at work in Bertrand Russell’s Religion and Science.* In the series’ previous posts, I have directed attention to Russell’s “exclusivist epistemological scientism,” as I have dubbed it, the thesis that scientific knowledge is the only knowledge. I have also taken note that an adoption of that thesis requires the adoption of the further theses that there is no theological knowledge distinct from scientific knowledge, no philosophical knowledge distinct from scientific knowledge, and no mathematical knowledge distinct from scientific knowledge. In this post, I wish to direct attention to one of the at least two complications that any understanding of Russell’s exclusivist epistemological scientism has to recognize: for the Russell of Religion and Science, there is not even scientific knowledge. The other of the two complications will have to await its share of attention until the next post.

2. The most salient passage in which Russell denies that scientific theory counts in fact as knowledge begins with his saying (p. 14),

A religious creed differs from a scientific theory in claiming to embody eternal and absolutely certain truth, whereas science is always tentative, expecting that modifications in its present theories will sooner or later be found necessary, and aware that its method is one which is logically incapable of arriving at a complete and final demonstration.

Russell proceeds then to tell us that science achieves (1) not “absolute truth,” but rather “practical truth” or “‘technical’ truth” and (2) not “knowledge,” but “‘knowledge’.” He first (pp. 14-15) sets absolute truth aside.

But in an advanced science the changes are generally only such as serve to give slightly greater accuracy; the old theories remain serviceable where only rough approximations are concerned, but are found to fail when some new minuteness of observation becomes possible. Moreover, the technical inventions suggested by the old theories remain as evidence that they had a kind of practical truth up to a point. Science thus encourages abandonment of the search for absolute truth, and the substitution of what may be called “technical” truth, which belongs to any theory that can be successfully employed in inventions or in predicting the future. “Technical” truth is a matter of degree: a theory from which more successful inventions and predictions spring is truer than one which gives rise to fewer.

Russell proceeds next (p. 15) to set aside scientific knowledge, leaving us with but scientific “knowledge.”

“Knowledge” ceases to be a mental mirror of the universe, and becomes merely a practical tool in the manipulation of matter. But these implications of scientific method were not visible to the pioneers of science, who, though they practiced a new method of pursuing truth, still conceived truth itself as absolutely as did their theological opponents.

3. a. Two things may be said on Russell’s behalf here. The first is that he is, of course, absolutely right in pointing to the real progress that has been made in scientific knowledge. For example, it was once believed by most, including Aristotle and even Descartes, that light traveled instantaneously. As Professor Michael Fowler of the University of Virginia notes in his “The Speed of Light,” Galileo, in his Two New Sciences, has his character, Simplicio, “stating the Aristotelian position,”

SIMP. Everyday experience shows that the propagation of light is instantaneous; for when we see a piece of artillery fired at great distance, the flash reaches our eyes without lapse of time; but the sound reaches the ear only after a noticeable interval.

Now, of course, we know that, and I’ll put it cautiously, in at least some circumstances, light travels at a finite velocity, and so with a “lapse of time.”

3b. The second thing that may be said on Russell’s behalf is that the knowledge that we thought we had, that, say, in all circumstances,** light travels instantaneously, turns out to have been, not knowledge at all, let alone absolute knowledge, but merely, using Russell’s way of putting it, “knowledge.”

4. It remains the case. however, that, in at least some circumstances, light travels at a finite velocity, and so with a “lapse of time.”***

It is the case, moreover, that the proposition, “In at least some circumstances, light travels at a finite velocity, and so with a ‘lapse of time,’” is a true proposition and not at all a false proposition. It will not be displaced by a “truer” proposition at any point in the future, thereby becoming less true.

And it is the case, finally, that we know that, in at least some circumstances, light travels at a finite velocity, and so with a “lapse of time.” That knowledge is a genuine knowledge and not at all a case of non-knowledge or mere “knowledge.” It will not be displaced by an opinion that is more a knowledge at any point in the future, thereby becoming less a knowledge.

5. Let me add hastily that the piece of reasoning just concluded does not commit us to the view that, though “knowledge” may not “be a mental mirror of the universe,” knowledge is such “a mental mirror of the universe.” The relation in which a knowing mind stands to that which it knows is a topic that will have to await a post that will appear, if at all, sometime in a perhaps distant future.

6. To sum things up temporarily: the Russell of Religion and Science, then, is committed on the one hand to the view that neither theology, nor philosophy, nor mathematics provides us with knowledge, though science does. On the other hand, he is also, and inconsistently, committed to the view that not even science provides us with knowledge, that is, to a thorough skepticism. I say, “temporarily,” because we have yet to take into account the second of the complications alluded to in the opening paragraph of this post. That we will do with the next post in the series.

7. I’ll conclude this post by recommending that you watch and listen to a comedic turning of Russell’s contrast between religion and science on its head, from the “Science is a liar…Sometimes” episode of the television series, It’s Always Sunny in Philadelphia.

Until next time.

Richard

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* Bertrand Russell, Religion and Science, with an introduction by Michael Ruse (Oxford and New York: Oxford University Press, 1997 [1935]). Religion and Science is readily available for purchase through Amazon.com. You need only click on the following image to be taken to the Amazon site:

** Though I’ve not researched the literature, I think it safe to assume that the vast majority of the statements of the view that light travels “instantaneously” left the “in all circumstances” clause unexpressed.

*** Our knowledge that, in at least some circumstances, light travels at a finite velocity, and so with a “lapse of time,” in no way rules out the at least logical possibility that we will at some point come to know that, in some other circumstances, light travels at an infinite velocity, and so with no “lapse of time.”

## Reflecting on Russell’s Religion and Science 2. Its Scientism Confirmed and Two Complications Raised

1. The present post is the second in a series of posts reflecting on the philosophical theses at work in Bertrand Russell’s Religion and Science.* In the series’ opening post, I did three things pertinent to the present one. First, I directed the readers’ attention to the opening paragraph of Russell’s essay, “The Art of Rational Conjecture, the first of the three essays that constitute the book, The Art of Philosophizing and Other Essays.** There (p. 1) Russell “gave expression … in a statement than which none more terse is possible” of the thesis of, as I dubbed it, “exclusivist epistemological scientism.” (If it is not immediately evident just what is meant by “exclusivist epistemological scientism,” a perusal of the previous post should help.). Russell’s statement reads:

Let us begin with a few words as to what philosophy is. It is not definite knowledge, for that is science. Nor is it groundless credulity, such as that of savages [sic]. It is something between those two extremes; perhaps it might be called the art of rational conjecture.

I took that identification of science with “definite” knowledge as a statement of the defining thesis of exclusivist epistemological scientism.

Second, over two or three steps, I expanded that statement of the defining thesis of exclusivist epistemological scientism into the fully explicit universal affirmative categorical proposition,

All instances of knowledge are instances of scientific knowledge.

and then converted that proposition into the logically equivalent,

Only instances of scientific knowledge are instances of knowledge.

Third, I set out as an historical task that of determining whether and, if so, to what extent Russell adheres to the doctrine of exclusivist epistemological scientism in Religion and Science. The aim of the present post is to complete that task.

2. The task is not that difficult to complete and will be quickly taken care of in the paragraphs to follow. There are, however, at least two complications; they will have to await a subsequent post or two.

Three texts demonstrate the presence of the doctrine of exclusivist epistemological scientism in Religion and Science. Its most full-throated expression is located at the end of the ninth chapter, “Science and Ethics.” There (p. 243) Russell tells us:

I conclude that, while it is true that science cannot decide questions of value, that is because they cannot be intellectually decided at all, and lie outside the realm of truth and falsehood. Whatever knowledge is attainable must be attained by scientific methods; and what science cannot discover, mankind [sic] cannot know.

There are two radical theses in the theory of ethics expressed here, the so-called emotivist theory of ethics. The one is that, as “questions of value” “lie outside the realm of truth and falsehood,”

No propositions of ethics are either true or false.

The other thesis is the thesis that as “questions of value” “cannot be intellectually decided at all,”

There is no ethical knowledge.

(The two theses are distinct the one from the other, for it is at least logically possible for a proposition to be true or false even though it knowing whether it is true or false may be beyond human capacity.)

That aside set aside, we can note that the proposition, “There is no ethical knowledge,” has to be accepted by one who, like Russell, accepts the theses that

All instances of knowledge are instances of scientific knowledge.

and

No instances of scientific knowledge are instances of ethical knowledge.

for together the two serve as the premises of the following patently valid argument (in Celarent; see the aforementioned immediately preceding post):

No instances of scientific knowledge are instances of ethical knowledge.
All instances of knowledge are instances of scientific knowledge.
Therefore, no instances of knowledge are instances of ethical knowledge.

3. Two further texts endorsing the doctrine of exclusivist epistemological scientism will catch the attention of readers of Religion and Science. In Chapter VI, “Determinism,” Russell tells us (pp. 144-145:

[T]here are three central doctrines—God, immortality, and freedom—which are felt to constitute what is of most importance to Christianity, insofar as it is not concerned with historical events. These doctrines belong to what is called “natural religion”, in the opinion of Thomas Aquinas and of many modern philosophers, they can be proved to be true without the help of revelation, by means of human reason alone. It is therefore important to inquire what science has to say as regards these three doctrines. My own belief is that science cannot either prove or disprove them at present, and that no method outside of science exists for proving or disproving anything.

And in Chapter VII, “Mysticism” (p. 189):

I cannot admit any method of arriving at truth except that of science….

4. In sum, the Russell of Religion and Science does not accept as real any knowledge other than scientific knowledge. Insofar, therefore, as he remains consistent with the doctrine of exclusivist epistemological scientism, he will have to eliminate the other three, mathematics, philosophy, and theology, of the four major theoretical disciplines or magisteria, as I called them in the previous post, leaving only science. But there remain the complications I mentioned above. I will turn my attention to them in the next post or two.

Until next time.

Richard

* Bertrand Russell, Religion and Science, with an introduction by Michael Ruse (Oxford and New York: Oxford University Press, 1997 [1935]). Religion and Science is readily available for purchase through Amazon.com. You need only click on the following image to be taken to the Amazon site:

*** Bertrand Russell, The Art of Philosophizing and Other Essays (Totowa, New Jersey: Littlefield, Adams & Co., 1974 [1968]). I continue to be puzzled over the fact that, in a book bearing such a title, there is to be found no essay entitled “The Art of Philosophizing.”

The Art of Philosophizing and Other Essays too is readily available for purchase through Amazon.com. You need only click on the following image to be taken to the Amazon site:

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