On the 5-point scale current in Estonia, and surely in nearby nations, and familiar to observers of the academic arrangements of the late, unlamented, Union of Soviet Socialist Republics (applying the easy and lax standards Kmo deploys in his grubby imaginary "Aleksandr Stepanovitsh Popovi nimeline sangarliku raadio instituut" (the "Alexandr Stepanovitch Popov Institute of Heroic Radio") and his grubby imaginary "Nikolai Ivanovitsh Lobatshevski nimeline sotsalitsliku matemaatika instituut" (the "Nicolai Ivanovich Lobachevsky Institute of Socialist Mathematics") - where, on the lax and easy grading philosophy of the twin Institutes, 1/5 is "epic fail", 2/5 is "failure not so disastrous as to be epic", 3'5 is "mediocre pass", 4.5 is "good", and 5/5 is "excellent"): 4/5. Justification: Kmo has time to make the necessary points to adequate length.
Revision history:
- 20161004T1444Z/version 1.2.0: Kmo added several sentences to the existing Moise quotation. (He had made the mistake of quoting too briefly.) - He reserved he right to upload minor (i.e., cosmetic, as opposed to substantive) tweaks over the coming 48 hours, as here-undocumented versions 1.2.1, 1.2.2, 1.2.3, ... .
- 20161004T0021Z/version 1.1.0: Kmo finished the final Moise quotation, and uploaded again, without at this point pausing to tweak his writing in cosmetic ways. He reserved the right to upload minor (i.e., cosmetic, as opposed to substantive) tweaks over the coming 48 hours, as here-undocumented versions 1.1.1, 1.1.2, 1.1.3, ... .
- 20161004T0001Z/version 1.0.0: Kmo uploaded most of the base version, while under time pressure leaving a bit of the final Moise quotation incomplete. He hoped to finish the quotation off in the next hour.
[CAUTION: A bug in the blogger software has in some past weeks shown
a propensity to insert inappropriate whitespace at some late points in
some of my posted essays. If a screen seems to end in empty space, keep
scrolling down. The end of the posting is not reached until the usual
blogger "Posted by Toomas (Tom) Karmo at" appears.]
3. Points of Detail from Moise
3.1 Formalism, Outside the Strict Domain of Logic
/.../
Moise, on the other hand, fills this
in with care, taking fully four pages, with a formal inductive proof,
and in the course of his work answering that very necessary question -
I imagine it said at the samovar in my imaginary, grubby, Nikolai
Ivanovitch Lobachevsky Institute of Socialist Mathematics, on the
scarier side of the Urals - "Vot MEENZ 'freely associative'?". He
starts his four-page discussion as follows (the emphasis on the word
"pairs" is his):
In practice /.../, as soon as you get past Chapter 1 of anybody's book, you are writing n-fold sums
a1 + a2 + ... + an,
and n-fold products
a1 a2 ... an,
for n > 3. We insert and delete parentheses in these sums and
products, at will. All this is fine, but it has not been connected up,
so far, with the operations that are supposed to be given for pairs
of numbers (a,b) and with the associate laws for triplets (a, b, c). It
would be a pity if mathematics appeared to be split down the middle,
with the postulates and definitions on one side, and the mathematical
content on the other. let us therefore bridge the gap between our
postulates and the things that we intend to do.
3.2 Formalism, Within the Strict Domain of Logic
A mild criticism, regarding the present King of France: Thirty-odd years ago, everyone in Canada knew two things exceptionally well, both of them from public broadcasting. One was "Hockey Night in Canada". The other was the evening CBC news, anchored from 1978 to 1988 by Knowlton Nash (1927-2014). Upon your applying power to the television, a sepulchral voice would announce, in the tones of high earnestness rather emblematic of Canada, "the ... NATIONAL ... with .. KNOWLTON NASH".
I think I was the one person living in, or as in my then case frequently visiting, Canada to find this comic.
Similarly comic is the history of analytical philosophy, and here again I appear alone in my amusement.
Analytical philosophy is the creation of the pre-First-World-War Anglo-Saxons - drawing, however, some key inspirations from the slightly earlier German mathematical logician Gottlob Frege (1848-1925).
The two key founding figures in this (on the whole worthy) movement, which (on the whole rightly) stresses the rigorous logical analysis of language, are G.E.Moore (1973-1958) and the 3rd Earl Russell (1872-1970). Their twin contributions to linguistic analysis inspired three or four generations, and no doubt continue to inspire. I, too, on the whole admire them. And yet they do to my irreverent temperament seem occasionally a little comic.
On the one hand we have Moore, analyzing "good" in his 1903 Principia Ethica. On the other hand we have Bertrand Russell's analysis, in "On Denoting" (Mind 14 (1905)), of a still shorter word, "the".
Moore's contribution was to argue that it is naughty to try defining the (indefinable) word "good". If you offer a definition, he argued, you fall into the "Naturalistic Fallacy": if to be good is to be a foobarzoogar (for instance, to be an action-tending-to-generate "utilitarian", i.e., Jeremy-Benthamite, pleasures), then the question arises, "But are foobarzoogars actually GOOD?" This question, says Moore, cannot be merely tautologous, as our question would be if we were so impossibly dim as to ask, "But are good things actually GOOD?"
Lord Russell's contribution was to analyze (no, I am not making this up, folks: read any history of modern philosophy) "The present King of France is bald."
Lord Russell's 1905 paper is perhaps not as clear as it could be. To improve a little on his own formulation, I put his point thus: "the present King of France" is not a proper name, such as "de Gaulle" or "l'Inspecteur Clouseau", but an expression built up with a binary quantifier, "the one and only thing which is such that ... is such that ... ." In this I follow one or more expositors of Russell, perhaps including logician Arthur Prior (1914-1969).
Lord Russell was at any rate well aware of the need to treat with care the scope of such a quantifier. It is one thing to say "The present King of France is such that every blogger who has shaken hands with him is tall" and a different thing to say "Every blogger who has shaken hands with the present King of France is tall." The first of these propositions might be formally expressed, with binary quantifiers, as
(1) The one and only entity v which is such that
v is a present King of France
is such that
every entity w which is such that
w is a blogger who has shaken hands with v
is such that
w is tall
The second might be expressed with binary quantifiers as
(2) Every entity x which is such that
both x is a blogger
and the one and only entity y which is such that
y is a present King of France
is such that
x has shaken hands with y
is such that
x is tall.
Since there is no present King of France, (1) cannot be true. (2), on the other hand, is true (at any rate on the treatment of universal quantifiers most convenient in mathematics, as when one is scrutinizing Edwin Moise), for the same reason that "Every integer in the empty set is odd" is true and "Every integer in the empty set is even" is true, or "Every real-number solution of 'x times x is -1' is an integral multiple of pi is true", or indeed "Every married spinster is blonde" is true. (On the most natural treatment of the universal quantifiers needed in mathematics, "Every entity which S-eth P-eth" is true in all and only those states of affairs that make "No entity which S-eth refrains from P-ing" true.)
Similar points arise (as Russell was aware) in the opposite situation, in which "The only and only entity which S-eth P-eth" gets into trouble because there is more than one S-ing entity. Since there is more than one present Canadian bishop, "Every entity which is such as to be a biographer of the present Canadian bishop speaks Danish" is vacuously true (just as "Every dragon speaks Danish," with the mathematicians' "every", is vacuously true), whereas "The present Canadian bishop is such that every entity which is a biographer of that bishop speaks Danish" cannot be true.
This leads, alas, to a mild criticism of Moise (whom I cite in the remainder of this essay exclusively from his third, i.e., final (1990) edition).
Moise (p. 44) lays down as a postulate, in what is intuitively speaking a setting of three-dimensional possibly-Euclidean-and-possibly-Lobachevskian-or-even-in-some-other-way-non-Euclidean space, "Given any two different points, there is exactly one line containing them." So far, so good. But now the ice gets a little thin. For Moise's next move is to introduce the notation "PQ", with a bidirectional (i.e., a two-headed) arrow over his pair of letters, as a symbol denoting "the line containing" points P and Q. Moise does not, alas, explain what sense is to be attached to sentences containing this "the line containing points A and B" notation in cases in which A and B turn out to be one and the same point (and in which, far from it being the case that one can point to "the line", one is therefore confronted by a plurality of lines).
Moise (p. 44) lays down as a postulate, in what is intuitively speaking a setting of three-dimensional possibly-Euclidean-and-possibly-Lobachevskian-or-even-in-some-other-way-non-Euclidean space, "Given any two different points, there is exactly one line containing them." So far, so good. But now the ice gets a little thin. For Moise's next move is to introduce the notation "PQ", with a bidirectional (i.e., a two-headed) arrow over his pair of letters, as a symbol denoting "the line containing" points P and Q. Moise does not, alas, explain what sense is to be attached to sentences containing this "the line containing points A and B" notation in cases in which A and B turn out to be one and the same point (and in which, far from it being the case that one can point to "the line", one is therefore confronted by a plurality of lines).
Of course we must try to take care, in using the notation, to ensure that A and B are distinct, just as we just take care in cases when writing the quotient "y/x", in the usual convenient-and-yet-occasionally-dangerous notation, to ensure that x is nonzero. Still, what happens **IF** A for some reason turns out to be the same point as B? To render the whole intellectual structure rainproof, we must have an answer (be it ever so contrived, be it ever so pedantic) to this question.
Since for Moise a line is a set of points, one might be tempted to rule that "the line containing points C and C" is the empty set of points. In this case, however, significant slabs of Moise would have to be rewritten, since Moise also has as a postulate "Every line contains at least two points."
Similar problems arise for planes. Moise postulates, acceptably, "Given any three different noncollinear points, there is exactly one plane containing them." He then, however, introduces a notation for "the plane containing points P, Q, and R" without legislating on what to do with "the plane containing points P, P, and Q", "the plane containing points P, Q, and P", "the plane containing points P, Q, and Q", "the plane containing points P, P, and P", and (where P, Q, and R are distinct-yet-collinear) "the plane contining points P, Q, and R" (all cases in which, far from it being the case that one can point to "the plane", one is confronted by a plurality of planes).
Trying to render the structure rainproof, I tentatively follow Lord Russell, tentatively thinking his solution to be the tidiest of the various available options. Lord Russell analyzed "The one and only entity t such that t is a present King of France is such that t is bald" as "At least one entity s is a present King of France, and there are not two distinct entities r and r' such that r is a present King of France and r' is a present King of France, and s is bald." This analysis has the consequence that "The present King of France is bald" (in this case, alas, no present King of France exists) and indeed also "The present Canadian bishop is bald" (in this case, alas, there is more than one present Canadian bishop), are false. - So too, then, I tentatively legislate that sentences containing such phrases as "the line containing A and B" be understood in terms of a Russellian binary quantifier "the one and only entity which is such that ... is such that...", with due attention paid to quantifier scope, and with the quantifier's truth-conditions given as Lord Russell gives them.
This tentative legislation has the consequence that Moise's postulates (notably, "Every line contains at least two points") can be allowed to stand unmodified, and that we know what to do with any constructions involving "the line containing A and A", should they ever arise - once quantifier scopes have been duly declared, we will be able to work out whether the embedding proposition is true or false.
As for "the line", so too for "the plane".
As far as I can see at present, no resort to Lord Russell beyond these two is necessary in the mildly tedious rainproofing exercise. In particular:
- As far as I can see at present, we can without any special awkwardness decree that if point A is the same entity as point B, then "the segment between A and B" (Moise fails to discuss its status) is that special set of points which is the empty set.
- As far as I can see at present, we can without any special awkwardness decree that if point A is the same entity as point B, then "the ray from A through B") (Moise fails to discuss its status) is that special set of points which is the empty set.
- As far as I can see at present, we can without any special awkwardness decree that if either points A, B, and C are not all distinct, or points A, B, and C, while distinct, nevertheless prove so ill-bred as to be collinear, then "the angle ABC" (Moise fails to discuss its status) is that special set of points which is the empty set.
Regarding angles, Moise carefully refrains from introducing "straight angles", "zero angles", and "reflex angles" (i.e., angles whose degree measure is strictly greater than 180 and strictly less than 360), and additionally carefully refrains from introducing such things as an "angle whose degree measure is 361" or an "angle whose degree measure is 7892" or an "angle whose degree meausre is -45". Such things are needed, admittedly, elsewhere, in trigonometry as opposed to geometry, and then are best introduced in the specialized framework of "directed angles" (or, equivalently, of "winding numbers"), as opposed to mere "angles". For Moise, an "angle" is, in his trigonometry-free framework, a pleasantly simple thing: a pair of distinct rays, possessing a common point of origin, whose union is not a line.
****
Moise's sensitive discussion of "converse": I had briefly to invoke the idea of of converse in my upload of "part A" from this essay, on 2016-08-15 or 2016-08-16. As I noted there, Euclidean geometry offers on the one hand the Pythagorean Theorem, and on the other hand its converse:
- if a triangle is a right triangle, the area of a square erected on the side opposite a dominant angle equals the sum of the areas of the squares erected on the other two sides;
- only if a triangle is a right triangle does the area of a square erected on the side opposite a dominant angle equal the sum of the areas of the squares erected on the other two sides
- where by "dominant angle" I mean "any angle of the given triangle than which there is none greater in
that triangle" (so that every triangle which is
isosceles-and-not-equilateral has either exactly two dominant angles or
exactly one dominant angle, and every equilateral triangle has three
dominant angles, and every triangle outside the two just-cited special
cases has exactly one dominant angle).
But now, having read a little further in Moise, I notice a small point, indicative of Moise's general excellence.
Despite teaching formal logic for a number of years in a number of different Departments of Philosophy, I seem to have failed to note the inadvisability of speaking of "THE" converse of a hypothetical proposition (or of the universal quantification of a hypothetical propositional function). For the working mathematician - Moise helpfully reminds those of us who aspire, however feebly, to join the community of mathematicians, in however tiny a capacity - it is more convenient to speak of "**A** converse" than to stick in pedantry to "**THE** converse". Suppose, as part of work-in-progress, we find that if both p and q then r. We at this point potentially face not one, but three research questions, perhaps all requiring investigation. It is only the first that would be called "the" converse of "If both p and q then r", on my old, narrow and pedantic, use of "converse":
- Is it true that if r then both p and q?
- Is it true that if both p and r then q?
- Is it true that if both q and r then p?
Moise's sensitive discussion of "identical": I have this past northern-hemisphere summer pointed out (in "Part A" of this essay, on 2016-08-15 or 2016-08-16) how Godrey and Siddons, in their unsatisfactory 1919 school textbook, write in unclear terms on identity or "equality", and have today noted how the concept of identity assumes a certain prominence upon analyzing "The line through A and A is so-and-so," "The plane through P, P, and Q is so-and-so," and the like (problematic propositions which, I have said, parallel "The present Canadian bishop is bald" - there are multiple lines through A and A, and multiple planes through P, P, and Q, just as there are multiple present Canadian bishops).
Moise is for his part correctly careful with identity. To illustrate this, it suffices to quote him verbatim (from his pp. 111-112: the emphases or italicizations are in Moise himself, but are here reproduced with underlining; and I paraphrase Moise's mathematical typesetting with the limited safe resources of the available blogging software, writing my paraphrases in square brackets):
There are two difficulties with the loose use of the word equals to describe equality of length, angular measure, and area. The first difficulty is that if the word equals is used in this way, there is no word left in the language with which we can say that A is - without ifs, buts, qualification, or fudging - the same as B. This latter relation is called the logical identity. It may seem a peculiar idea, at first, because if two things are exactly the same, there can't be two of them. But as soon as mathematics had begun to make heavy use of symbolism, the logical identity became important. For example, each of the expression
[1/(2 (nonnegative-square-root-of 3) - 1)], [(2 (nonnegative-square-root-of 3) +1) /11]
describes a number. The descriptions are obviously different, but it is easy to check that they describe the same numer,; and this is what we mean when we write
[1/(2 (nonnegative-square-root-of 3) - 1) = (2 (nonnegative-square-root-of 3) +1) /11]
The relation denoted by the symbol =, in the above equation, is the logical identity.
The concept of the logical identity A=B is so important, and comes up so often, that it is entitled to have a word to itself. For this reason, in nearly all modern mathematics, the word equals and the symbol = are used in only one sense: they mean is exactly the same as.
The second difficulty with the loose use of the word equals is that it puts us in the position of using two words to describe the same idea, when one word would do. Congruence is the basic equivalence relation in geometry. we may use different technical definitions for it, in connection with different types of figures, but the underlying idea is always the same: two figures are congruent if one can be placed on the other by a rigid motion. The basic equivalence relation of geometry is entitled to hae a word all to itself; and the word congruence appears to be elected.
It was, no doubt, for these reasons that Hilbert adopted, in his Foundations of Geometry, the terminology that we are using in the present book. The problem is not logical but expository. A good terminology matches up the words with the ideas in the simplest possible way, so that the basic words are in one-to-one correspondence with the basic ideas.
It should be borne in mind that the strict mathematical interpretation of the word equals, in the sense of "is exactly the same as," is a technical usage. In ordinary literary English, the word is used even more loosely than in Euclid. For example, when Thomas Jefferson wrote, in the Declaration of Independence, that all men are created equal, he did not mean that there is only one man in the world, or that all men are congruent replicas of one another. He meant merely that all men have a certain property in common, namely, the property of being endowed by their Creator with certain unalienable rights.
In fact, only mathematics and logic need a word and a symbol for the idea of "is the same as," and the need did not appear even in mathematics and logic until the heavy use of symbolism developed. This development came long after Euclid; and sheer force of habit preserve Euclid's terminology long past the time when it had become awkward.
Moise is for his part correctly careful with identity. To illustrate this, it suffices to quote him verbatim (from his pp. 111-112: the emphases or italicizations are in Moise himself, but are here reproduced with underlining; and I paraphrase Moise's mathematical typesetting with the limited safe resources of the available blogging software, writing my paraphrases in square brackets):
There are two difficulties with the loose use of the word equals to describe equality of length, angular measure, and area. The first difficulty is that if the word equals is used in this way, there is no word left in the language with which we can say that A is - without ifs, buts, qualification, or fudging - the same as B. This latter relation is called the logical identity. It may seem a peculiar idea, at first, because if two things are exactly the same, there can't be two of them. But as soon as mathematics had begun to make heavy use of symbolism, the logical identity became important. For example, each of the expression
[1/(2 (nonnegative-square-root-of 3) - 1)], [(2 (nonnegative-square-root-of 3) +1) /11]
describes a number. The descriptions are obviously different, but it is easy to check that they describe the same numer,; and this is what we mean when we write
[1/(2 (nonnegative-square-root-of 3) - 1) = (2 (nonnegative-square-root-of 3) +1) /11]
The relation denoted by the symbol =, in the above equation, is the logical identity.
The concept of the logical identity A=B is so important, and comes up so often, that it is entitled to have a word to itself. For this reason, in nearly all modern mathematics, the word equals and the symbol = are used in only one sense: they mean is exactly the same as.
The second difficulty with the loose use of the word equals is that it puts us in the position of using two words to describe the same idea, when one word would do. Congruence is the basic equivalence relation in geometry. we may use different technical definitions for it, in connection with different types of figures, but the underlying idea is always the same: two figures are congruent if one can be placed on the other by a rigid motion. The basic equivalence relation of geometry is entitled to hae a word all to itself; and the word congruence appears to be elected.
It was, no doubt, for these reasons that Hilbert adopted, in his Foundations of Geometry, the terminology that we are using in the present book. The problem is not logical but expository. A good terminology matches up the words with the ideas in the simplest possible way, so that the basic words are in one-to-one correspondence with the basic ideas.
It should be borne in mind that the strict mathematical interpretation of the word equals, in the sense of "is exactly the same as," is a technical usage. In ordinary literary English, the word is used even more loosely than in Euclid. For example, when Thomas Jefferson wrote, in the Declaration of Independence, that all men are created equal, he did not mean that there is only one man in the world, or that all men are congruent replicas of one another. He meant merely that all men have a certain property in common, namely, the property of being endowed by their Creator with certain unalienable rights.
In fact, only mathematics and logic need a word and a symbol for the idea of "is the same as," and the need did not appear even in mathematics and logic until the heavy use of symbolism developed. This development came long after Euclid; and sheer force of habit preserve Euclid's terminology long past the time when it had become awkward.
[To be continued and concluded next week, as Part F?]
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