The Source of Mathematical Truth is Ineffable
I think it is time to reveal why Triadic Philosophy takes the position that some of Reality (which is All) is Mystery. It is because that is the case.
But it is also the case that some things that are mystery are that way because they are operational. They are under our noses but we cannot explain them. Oh, we try. But ineffabillity kicks in.
Which means that the various efforts to suggest that mathematical truth can be expressed in syllogisms is a stretch. If you look below, you will see a sure-footed effort to arrive at a proper logical expression.
Peirce on the trail!
But once we get to his rest stop, we arrive not at truth but supposition. And that is where we mostly arrive everywhere we go. So when we say that logic is goodness, we are inferring. And inferring is educated guesswork.
But it is as close as we may get and to achieve goodness we need something to base inference on and that something in Triadic Philosophy is values which carry with them the aura of truth and beauty, not to mention universality and ontological status.
I live in a mathematical jungle called Manhattan. It is so because its buildings are the product of logic derived from mathematics.
This logic has not operated autonomously. It has been used, utilized, employed and run by a debased form of logic which I do not hesitate to call real estate interests.
And these are run by all manner of powers and interests.
And the result is not logic that is goodness, but logic far down on the scale of values. Logic in the deterioration realm. Logic where mindlessness and selfishness. Logic that justifies corruption and criminality.
Logic is goodness and that means that the thought that underlies even a seemingly flawless syllogism is subject to a sort of Blakean examination.
Life is simply unable to operate without the absurdity to which Peirce refers.
Buildings may be created flawlessly on the basis of mathematical formulae and slide rule figuring, But their place in reality is not thereby a paean to logic. It celebrates a debased conception that exists because values were not considered when the building was conceived and built.
When values are not understood or conceived, the lesser values kick in.
Now you may say that those stately skyscrapers old and new are a mixed bag but surely if they are emulated worldwide they are a mixture, not an unalloyed evil. And that is of course true. My only response as always is to refer to what is no more than a draft of what seems to me to be the fundamental chart the world needs to begin putting itself into some logical mode of existence.
Peirce: CP 2.77 Cross-Ref:††
77. What then is the source of mathematical truth? For that has been one of the most vexed of questions. I intend to devote an early chapter of this book to it.†1 I will merely state here that my conclusion agrees substantially with Lange’s, that mathematical truth is derived from observation of creations of our own visual imagination, which we may set down on paper in form of diagrams. When it comes to logical truth, I do not think the intuition is quite what Lange describes it. He holds that we imagine something like an Euler’s diagram; but I do not think that necessary. There are other ways, as I shall show, among which we may take our choice.†2 Lange holds up as a model Aristotle’s proof of the conversion of the universal negative proposition. As well as I can translate Aristotle’s untranslatable language, his proof reads as follows:
Peirce: CP 2.77 Cross-Ref:††
“If to none of the B’s the (designation) A belongs; neither will the (designation) B belong to any of the A’s. For if to any, as, for example, to C, it will not be true that to none of the B’s the (designation) A belongs. For C is one of the B’s.”†3
Peirce: CP 2.78 Cross-Ref:††
78. It seems to me it would be much simpler to say that if No B is A, but some A is B, then we should have the two premisses of a syllogism in Ferio, from which we could conclude: Some A is not A, which is absurd. Hence, if No B is A, it cannot be true that some A is B; that is, it must be true that No A is B. The syllogistic form Ferio
Some C is B,
.·. Some C is not A.
If for C we put A, we get the premisses above used. Now in my view, observation comes in to assure us that when A is substituted for C we do get these premisses, and it also enters in other similar ways.