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 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.
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.
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, together with God, 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).