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Two Necessities

A square tablecloth spread on a rectangular table will hang from the table edges same as before if the cloth is rotated ninety degrees lying on the table. A rectangular cloth not square—a cloth made longer in one length, shorter in the other—will require two rotations of ninety degrees to hang from the table edges same as before. We can open our geometry books and find that such facts are aspects of the rotational isometries of quadrilaterals.

Facts of pure geometry can have explanatory value to us for some facts about table cloths. I should distinguish two sorts of necessities in pure geometries. One sort is the plain necessity of geometric facts, such as one-hundred-eighty degrees being the sum of the angles in any triangle in the Euclidean plane or such as any square’s diagonal being not of any integer-ratio to to the length of its sides. The other sort is the necessity between if and then in the inferences one makes in proving such results.

The former sort of necessity in pure geometry is identical to that necessity in its instances in physical space, such as the space of table cloths. But the second sort of necessity in pure geometry, the if-then necessities certifying facts in pure geometry, is not that first sort of necessity and is not at hand in any necessary geometric facts of table cloths. The geometric necessities in physical facts or in their counterpart necessities in pure-geometry facts do not somehow arise from the if-then necessities in the proofs in pure geometry. The latter necessities, indeed, are irrelevant to the former ones of themselves, the fact-necessities of themselves. The if-then necessities were required for us to know the complete absence of contrary cases in all reality of these geometric fact-necessities, but if-then necessities are not the source of geometric fact-necessities.

What I’m calling fact-necessities are examples of the necessity that existence is and is some identity (likewise for Ayn Rand; ITOE App. 299). Those necessities are in the world apart from the existence of any consciousness of them. As Leonard Peikoff points out in “The Analytic-Synthetic Dichotomy” (1967), such necessities have often been characterized not as necessities, but as contingencies (ITOE 92, 106–11). This was because of the religious doctrines that God, by choice, design-engineered the operations of the material world. By such lights as Leibniz, many basic facts—in mathematics, logic, and morality—are coeternal with God and not amenable to God’s revision of them. Those would be the only fact-necessities in that tradition. God’s ability to choose is the root of what makes a fact not necessary, but contingent in that tradition. Similarly, for Peikoff and Rand, man’s ability to chose in his devisings is the root reason his productions are not absolutely necessary in their every aspect (ITOE 110; Rand 1973).

I should maintain, in some contrast to Objectivism and other philosophies, that contingency of facts is not rooted ultimately in choices of conscious beings, but in the ends-means character of living organizations of existence, natural or artificial, or in nonliving mechanisms and systems devised by conscious living organizations of existence. In sum fact-necessity belongs to all facts, not just the ones known as eternal truths (this point contra Leibniz and in agreement with Rand-Peikoff). Contingency as contrast to fact-necessity is nothing at all. Contingency that is something is only in operation of (i) living systems (including the consciously living) and (ii) engineered systems. Both are of course embedded in fact-necessities (which embedment and sourcing of contingency needs elaboration in future writings).

(The indeterminism [not mere uncertainty] in aspects of canonically conjugate dynamical variables in the quantum regime has the same fact-necessity as the full determinism of those pairs of variables in the classical regime. Determinism has some distinction from necessity.)

The distinction between fact-necessity and if-then necessity also applies to mathematics outside geometry. The fact that every odd counting number squared and having 1 subtracted from it is divisible by 4 (i.e., when divided by 4, yields zero or a positive counting number without remainder) has the necessity of fact. That necessity does not derive from the if-then necessity (specifically, mathematical induction in this example) we rely on in discovering this fact about every odd counting number.

 

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(The following remark is from a post I made on the "Peikoff's Dissertation" thread on 8 October 2019. I was trying to quote this post from that thread and land it in this one when I goofed up and accidentally deleted the entire PD thread. Greg is investigating if can be retrieved. If not, then I'll recreate that thread there in my BOOKS TO MIND sector from my word processing file of it, which is where the text below comes from.)

There are patterns of exclusion for which one knows that no counterexample could be found. Such excluded combinations are not in existence and cannot be in existence. Examples are:

[A] A counting number cannot be both even and odd.

[ B] The sum of interior angles of a triangle in a Euclidean plane cannot be any value but 2R.

[C] An oscillation of inconstant period cannot at the same time be both increasing and decreasing in frequency.

[D] There are various things; a thing cannot be both itself and a thing not itself.

All varieties of logical ontologists (excepting the coarse-weaver Mr. Locke) and all their opponents agree that those patterns of exclusion, as necessarily so, can be seen only through work of the intellect, not seen in simply sensory perception. How the mind renders this intellectual seeing and what is the relation of this seeing to sensory perception are the contested matters.

[ B] and [C] are facts entirely independently of the existence of mind in the world. [A] is a fact about facility of mind in its engagement with mind-independent numerosity: I’ve got five-fingered hands, but whether a particular finger is even or odd depends on a count and the order in which I include that finger in the count. Unlike in my post of . . . I now understand the second clause in [D], which is PNC, to be like [A], not like [ B] or [C]. The first clause in [D] is the mind-independent basis of PNC, and it is discerned by the senses as well as by the intellect. PNC requires that mind-independent fact, but also the mind activity of mapping. Counting and mapping have mind-independent facts to cognize, and these cognitions, of course, do not exist in the world without mind, specifically intellect.

Edited by Boydstun
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The progression of odd squares increase by a multiple of 8

32-12=8
52-32=16
72-52=24

By extension the  first, second, third feed back into the discovery that add more evidence in the from of 1(st)*8=8, 2(nd)*8=16, 3(rd)*8=24, taken in conjunction with 8/4=2, 16/4=4, 24/4=8 should lead any reasoning being to any multiple of 8 is divisible by 4 without remainder.

While it may be a known fact in the database of mathematics, it is an aspect previously unconsidered to me, yet was readily and easily verifiable.

The if-then necessity is not as readily apparent to this reader from the above. Such knowledge teases of being beneficial to the relationship between Geometric Dimensioning and Tolerancing as it relates to issues that can arise in commercial manufacturing concerns.

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The two necessities seem categorically disparate.  There is so little linking them in reality that using the same term “necessity” to refer to both seems like a side effect of history rather than some broad metaphysical integrative principle.

One seems to encapsulate how and what things are, the other deals with how a mind is to work if is to be successful at attaining Sophia of those things which are.

Perhaps this requires a bold and creative shift in vocabulary, to clearly differentiate between these two things.

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Greg,

The if-then necessity I’m talking about is the necessity of a conclusion being true if the premises are true. I mean the deductive if-thens, that is, ones in which that necessity is total. There are physical relations whose necessity from all we know seem quite necessary, as when we say “if a bird has talons, then it is a bird of prey.” The if-then necessity I’m focussed on in the present exploration is not of that sort, the sort we get from empirical inductions or from a consilience of a variety of empirical inductions coming from different angles on an issue.

An example of an if-then deduction in which the necessity is gotten by the necessity of noncontradiction would be proof by contraposition. An example would be: If x is a rational number and y is an irrational number, then their sum is an irrational number. Proof: Suppose that x + y is a rational number. On that supposition, one can construct a contradiction (I’ll show this if there is an interest voiced to see those particulars), showing that supposition to be false. Therefore, the sum must be an irrational number.

Noncontradiction is not our only tool for absolute conveyance of the truth of premises to truth of a conclusion. Another tool for such conveyance is mathematical induction (which is not empirical induction). The principle of mathematical induction (in an elementary type of it) is:

A statement P(n) to be proved for integers n greater-or-equal-to t, where t is a fixed positive integer, is true provided: (i) P(t) is true, and (ii) if P(t), P(t+1), . . . P(n) are true, then P(n+1) is true.

(This is not a vicious circle. It provides conclusive, deductive proof. It was not known to the Greeks, though Euclid used it implicitly at one point.)

First verify (i), that P(n) is true for n=t. Then try to show that assuming the antecedent of (ii) being true implies that the consequent of (ii) is also true. Then we’ve proven the statement P(n) true. Using this tool, mathematical induction, we can PROVE that for any and every odd counting number, that number squared minus 1 is evenly divisible by 4. (See page 46 of my “Induction on Identity” [1991]). Such a fact cannot be uncovered by inspection, by empirical induction, nor by contraposition.

SL,

I think your way of distinguishing the two could be right provided that we connect the two at least by having the if-then necessity resting on the fact-necessity, such as when we rest the rule of noncontradiction on the fact of identity in reality.

Edited by Boydstun
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