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.

I have, further, two hopes. The one, minimal, hope, is that I will have you as a reader and that you will follow the series as it unfolds. I invite you, then, if you have not already done so, to go to the bottom of this page’s right-hand panel to Follow Blog via Email and enter your email address.

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.

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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

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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 to 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.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 ~Ixy) [Pr., Def.]
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 ii ~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

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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.

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)))

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

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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.”

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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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Reflecting on Russell’s Religion and Science 1. Scientism and the Four Purported Magisteria

1. One reason why I have not been posting over the past months is because I have been caught up in the teaching of courses new to me, in ethics, medical ethics, and environmental ethics. These having been courses for which I had not previously gone through the course preparation process, I have had to devote enough of my time in course preparation and management that I have had none left to devote to my blog.

It recently occurred to me, however, that I could and should combine my course preparation and my blogging, by posting to my blog, if sometimes in somewhat variant versions, the material that I am covering in the courses I teach. As it happens, I am scheduled to teach an introduction to philosophy course later this summer and it seems to me that that course is perfectly well fitted for the kind of integrated activity I have in mind. I intend, therefore, over the coming weeks and months, to offer in the form of posts to this blog some reflections on the material I plan to cover in this summer’s introduction to philosophy course (the courses in ethics, medical ethics, and environmental ethics can await their turn).

2. The primary text of the course will be Bertrand Russell’s Religion and Science,* published in 1935. This is not the book of Russell, it seems to me, that is most likely to show up in an introduction to philosophy course. That would be his The Problems of Philosophy. Indeed, I judge the latter to be the better book of the two. I have, however, used it, the latter, as the primary text in an introductory course in the past and quickly learned that my students found its first chapter’s extended focus on that which we can know of the author’s writing table to be, well, off-putting.

Religion and Science it is, then. Given, now, the title of the work, its readers can justly hope that the book will set forth at least some of Russell’s views on religion and science and on the relationship or relationships that exist between them. Given further that the book is being assigned in a course intending to introduce students and others to philosophy, those reading it with me can also justly hope that the book will set forth at least some of Russell’s philosophical views, so adding philosophy and the relationships it may have to religion and science into the mix of matters to be pondered. Given yet further that those reading Religion and Science with me might well notice that the back cover of the edition of the work at hand identifies Russell as an “English philosopher and mathematician,” they might well also hope that Russell’s understanding of mathematics and the relationships it may have to religion, science, and philosophy will be the object of some attention.

The last hope, that Russell will have interesting things to say about mathematics in Religion and Science, may yield in the course of its reading to disappointment. But hoping that the book will have interesting and important, if not therefore always also true, things to say about religion, science, philosophy, and the relationships between and among them will, I can guarantee, not have been in vain. And I, on the other hand, hope that I will be able to make up for the lacunae in the treatment in Religion and Science of mathematics by bringing in on occasion what Russell has to say on the subject elsewhere, what others have had to say, and, passing, to be sure, from the more significant to the less, what I have to say.

3. And so one of the major themes of the introductory course at hand and of the accompanying series of posts will be the respective natures of the four purported magisteria of science, mathematics, philosophy, and religion. I owe, of course, the use of the term, “magisteria,” to Stephen Jay Gould and his theory of the “nonoverlapping magisteria,” typically abbreviated as “NOMA.” According to Gould’s “Nonoverlapping Magisteria,”** the “NOMA principle” holds that:

Science tries to document the factual character of the natural world, and to develop theories that coordinate and explain these facts. Religion, on the other hand, operates in the equally important, but utterly different, realm of human purposes, meanings, and values—subjects that the factual domain of science might illuminate, but can never resolve.

I should perhaps, and in the not too distant future, devote a post to a comparison of Gould’s views on the nature of science, that of religion, and the relationship or relationships between them, with Russell’s. For the purposes of the present post, however, I will let the opportunity for that comparison pass and content myself with the observation that there are not just the two purported magisteria that Gould had in mind, but four, existing in the following hierarchy very evident in, at least, classical Western thought:

4. Theology
3. Philosophy
2. Mathematics
1. Science

I have placed theology at the top of the hierarchy, instead of religion, because it is theology, as the theoretical component of the religions Russell and classical Western thought have had most in mind, that most appropriately stands in correspondence with the theoretical disciplines of science, mathematics, and philosophy.

4. Returning to Russell, I will propose to my students and fellow readers that our first focus be on his “exclusivist epistemological scientism” (as I will dub it; I will clarify the “exclusivist” and the “epistemological” presently). Russell gave expression to this view of the place of science in a statement, than which none more terse is possible, in the opening paragraph of the 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), in the course of telling us what he thinks philosophy is and is not, Russell also tells us what he thinks science is.

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.

The defining thesis of exclusivist epistemological scientism is, then, the thesis,

Knowledge is science.

or, expanding its statement a bit,

All knowledge is scientific knowledge.

or, expanding the statement yet further so that we have before us a fully explicit universal affirmative categorical proposition, as logicians are wont to identify it,

All instances of knowledge are instances of scientific knowledge.

or, equivalently, thus making more fully explicit the reason for the “exclusivist” of the “exclusivist epistemological scientism” tag,

Only instances of scientific knowledge are instances of knowledge.

5. This thesis is, of course, an epistemological thesis, in the jargon of philosophy; in a somewhat more ordinary language, it is a thesis in the philosophical theory of knowledge. It is moreover, to put it mildly, a radical thesis in the philosophical theory of knowledge. If, indeed, it is true that, to revert to its last-but-one formulation, all instances of knowledge are instances of scientific knowledge, then we are faced with a set of arguments, surely disconcerting at least to some or even to many, but also nonetheless absolutely valid arguments.

First,

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

That is, in brief, there is no, as some have purported there to be, theological knowledge.

Then,

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

That is, in brief, there is no, as some have purported there to be, philosophical knowledge.

And, finally,

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

That is, in brief, there is no, as some have purported there to be, mathematical knowledge.

We should note that the three arguments all exhibit the same logical structure, the one to which medieval logicians attached the name, “Celarent” (you’ll have to google it; this post is already too long). The arguments are, as already noted, all valid: that is, if, that is, if, their premises are true, the conclusions must also be true. I’ll leave unanswered, for now, the further question of whether any or all of the arguments are sound, that is, of whether, in addition to their being valid [correction, June 1, 2020, replacing “sound” with “valid”], they are such that, in each case, both of their premises are true.

6. Now, according to the “Publisher’s Preface” to The Art of Philosophizing and Other Essays, the three essays contained in “this little volume” and thus too the “The Art of Rational Conjecture,” “were written by Bertrand Russell during the second World War” and so at least several years, if not nearly a decade, after Religion and Science. We should set for ourselves the historical task, then, of determining whether and, if so, to what extent Russell adheres in the earlier work to the several theses just set forth. This we will do in the coming series of posts. More importantly, we should set for ourselves the further, philosophical, task of reflecting with and/or against Russell, and others, and determining for oneself which of the several theses, if any, might be true. This too we will do.

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:

** As of this writing, Gould’s “Nonoverlapping Magisteria,” can be read online at The Unofficial Stephen Jay Gould Archive, http://www.stephenjaygould.org/library/gould_noma.html.

*** Bertrand Russell, The Art of Philosophizing and Other Essays (Totowa, New Jersey: Littlefield, Adams & Co., 1974 [1968]). I admit to being 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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Non-contradiction, Divine Omnipresence, and Dual Citizenship

1. Good Aristotelian that I am, or at least neo-Aristotelian, I introduce and make use of the principle of non-contradiction, that

No being can both be and not be, in the same respect and at the same time.

in virtually every course I teach. One immediate application of that principle is the thesis that

No being can both be in any one place and not be in that same place, in the same respect and at the same time.

2. I have found that students sometimes have difficulty understanding the difference between that thesis and the thesis that

No being can both be in one place and be in another place, in the same respect and at the same time.

My attempts at getting them to see the difference have seldom gone much beyond (1) repeating the two theses slowly and (2) and then pointing out that the thesis of divine omnipresence of classical theology, that

At least one being, God, can both be in one place and be in another place, in fact, in all other places, in the same respect and at the same time.

is not the same as the thesis, contradicting the principle of non-contradiction, that

At least one being, God, can both be in one place and not be in that same place, in the same respect and at the same time.

One can, that is, consistently uphold both the principle of non-contradiction and the thesis of divine omnipresence.

3. I believe I have found a more helpful, because more concrete, first exemplification of the difference at hand, for another immediate application of the principle of non-contradiction is the thesis that

No person can both be a citizen of any one nation and not be a citizen of that same nation, in the same respect and at the same time.

I have found that students have little difficulty in understanding the difference between that thesis and the thesis that

No person can both be a citizen of any one nation and be a citizen of another nation, in the same respect and at the same time.

or its equivalent

At least one person can both be a citizen of any one nation and be a citizen of another nation, in the same respect and at the same time.

for they can see that, in fact, not just one person, but many persons are citizens of two nations; they hold dual citizenships.

In my efforts to make clear that the principle of non-contradiction does not rule out a being’s being in one place and in another, I will in the future bring in the possibility of dual citizenship before that of divine omnipresence.

Until next time.

Richard

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Michael Anton on Behalf of the Thesis That Religion Is the Basis of Republican Government

1. No doubt because of the political circumstances in which the United States currently finds itself, I have in turn found myself wanting to know more about and better understand the political theory and philosophy of the nation’s founders, the “founding fathers.” So, coming upon Michael Anton’s “Founding philosophy. A review of The Political Theory of the American Founding by Thomas G. West” in The New Criterion and noting that Anton had served as Deputy Assistant to President [Trump] for Strategic Communications, I thought that the review might well be worth the reading.

2. It was, and for one reason because it contains a relatively explicit argument on behalf of the thesis that religion is the basis of republican government, a thesis that has considerable currency these days. Anton’s expression of the argument is concise, stated almost in passing.

In reality, West shows that Jefferson—like all the founders—well understood that republican government is impossible absent a strong moral foundation in the people, which in turn depends on religion, which government therefore has a duty to promote.

3. Before, however, entering upon the task of “regimenting” the argument, i.e., of so restating it that its logic is more fully explicit, it is worth noting that Anton, in stating that Jefferson “well understood” the sequence of thoughts expressed in the sentence just quoted, is both saying that that sequence of thoughts represents Jefferson’s view and that he, Anton, himself is in agreement with Jefferson on the matter.

That said, let’s observe that the one sentence quoted above actually gives expression to two arguments, the second of which is represented solely by the “which government therefore has a duty to promote,” i.e., the conclusion that:

Republican government has a duty to promote religion.

I’ll return to that argument below, if only briefly.

4. The first of the one sentence’s two arguments is expressed in the passage’s “republican government is impossible absent a strong moral foundation in the people, which in turn depends on religion.” Anton’s statement can be so reformulated that the argument’s validity is patently obvious.

As a first step, the “republican government is impossible absent a strong moral foundation in the people” can be restated as:

Only peoples having a strong moral foundation are peoples [capable of] having a republican government.

Similarly, the “which [strong moral foundation in the people] in turn depends on religion” can be restated as:

Only peoples having religion are peoples having a strong moral foundation.

Together the two propositions serve as the premises of the following argument:

Only peoples having a strong moral foundation are peoples [capable of] having a republican government.
Only peoples having religion are peoples having a strong moral foundation.
Therefore, only peoples having religion are peoples [capable of] having a republican government.

5. It is evident that the argument is perfectly valid, in that, if its two premises are true, then its conclusion is also, and necessarily, true. Perhaps, however, it will be worth taking the few simple steps needed to make its validity even more evident. First, then, the latter of the two propositions just spelled out is logically equivalent to the universal categorical proposition that:

All peoples having a strong moral foundation are peoples having religion.

The former of the two propositions just spelled out is logically equivalent to the universal categorical proposition that:

All peoples [capable of] having a republican government are peoples having a strong moral foundation.

Together the two propositions serve as the premises of the following argument:

All peoples having a strong moral foundation are peoples having religion.
All peoples [capable of] having a republican government are peoples having a strong moral foundation.
Therefore, all peoples [capable of] having a republican government are peoples having religion.

Students of traditional logic will recognize that this argument is a categorial syllogism of the form known as Barbara, all instances of which are valid.

6. Anton’s argument on behalf of the thesis that religion is the basis of republican government, i.e., that all peoples [capable of] having a republican government are peoples having religion, is, as I said above, a valid argument. That, however, tells us only that, if the two premises are true, the conclusion is also true; it does not tell us that the conclusion is true, tout court. The argument can tell us that the conclusion is true only if it, the argument, is sound, only, that is, if both the argument is valid and its premises are true.

7. Our question, then, is that of whether the premises of Anton’s argument are both true. Now, of course, they are not “self-evident,” as, say, “all bachelors are unmarried males” is self-evident. They therefore can only be known to be true if they are conclusions of sound arguments.

The argument on behalf of the conclusion that

All peoples having a strong moral foundation are peoples having religion.

would have to be an argument similar in form to Anton’s argument and so read like this:

All peoples [having some requisite characteristic] are peoples having religion.
All peoples having a strong moral foundation are peoples [having some requisite characteristic].
Therefore, all peoples having a strong moral foundation are peoples having religion.

8. I have invited, or perhaps challenged,** Mr. Anton to provide such an argument. I also invite any reader to provide one. And, while I was and am at it, I further invited him and now also invite the reader to provide the premises that would, in a valid argument, conclude to the aforenoted thesis that:

Republican government has a duty to promote religion.

The thesis that “all peoples having a strong moral foundation are peoples having religion” will undoubtedly be a premise of the argument, but not the only premise.

Until next time.

Richard

* Michael Anton’s “Founding philosophy. A review of The Political Theory of the American Founding by Thomas G. West” (The New Criterion, Vol. 36, No. 10, June 2018).

** Or tried to invite or challenge; finding a working email address for people as much in the public eye as Mr. Anton is difficult.

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