Jon. List
For the thing, why statistics is not pure mathemathics, I can only imagine, that its subject randomness is something external to mathematics. Is it? Randomness a phenomenon from other sciences like phaneroscopy? I cannot totally disagree with that view.
Being valid for an "ideal state of things" means, that parameters are limited to a number so that you can calculate with them.
But in chaos theory this limit is crossed. Again, you may say, that this is an influence from other sciences, or from experience.
I have, from the philosophical standpoint about classification of sciences no ready-made opinion about that. But on the other hand I have the feeling, that randomness and non-predictability of the outcome of nonlinear functions are experiences you can make without referring to other sciences, just by dealing with (e.g. complex, as you, Robert, said) numbers. That would be without leaving the range of mathematics. Somebody else might say, that randomness and losing the overview due too many parameters are influences from outside mathematics. But from where? Reality? Mathematical objects are by definition (of "reality") parts of reality too.
Best
Helmut
14. Juli 2021 um 19:47 Uhr
"Jon Alan Schmidt" <jonalanschm...@gmail.com>
wrote:
"Jon Alan Schmidt" <jonalanschm...@gmail.com>
wrote:
Helmut, List:
Again, anyone is free to disagree with Peirce's definition of mathematics. However, the occurrence of surprises is perfectly consistent with its method being strictly deductive and its subject matter being strictly hypothetical, especially given the distinction that he draws between corollarial and theorematic reasoning.
CSP: Deductions are of two kinds, which I call corollarial and theorematic. The corollarial are those reasonings by which all corollaries and the majority of what are called theorems are deduced; the theorematic are those by which the major theorems are deduced. If you take the thesis of a corollary,--i.e. the proposition to be proved, and carefully analyze its meaning, by substituting for each term its definition, you will find that its truth follows, in a straightforward manner, from previous propositions similarly analyzed. But when it comes to proving a major theorem, you will very often find you have need of a lemma, which is a demonstrable proposition about something outside the subject of inquiry; and even if a lemma does not have to be demonstrated, it is necessary to introduce the definition of something which the thesis of the theorem does not contemplate. In the most remarkable cases, this is some abstraction; that is to say, a subject whose existence consists in some fact about other things. ...Deduction, of course, relates exclusively to an ideal state of things. A hypothesis presents such an ideal state of things, and asserts that it is the icon, or analogue of an experience. (CP 7.204-205, EP 2:96, 1901)
Moreover, it seems to me that Peirce would not consider the discipline of statistics to fall within pure mathematics. Instead, it is an application of mathematics and thus belongs to the mathematical parts of other sciences. As I quoted him yesterday ...
CSP: What the mathematicians mean by a "hypothesis" is a proposition imagined to be strictly true of an ideal state of things. In this sense, it is only about hypotheses that necessary reasoning has any application; for, in regard to the real world, we have no right to presume that any given intelligible proposition is true in absolute strictness. On the other hand, probable reasoning deals with the ordinary course of experience; now, nothing like a course of experience exists for ideal hypotheses. Hence to say that mathematics busies itself in drawing necessary conclusions, and to say that it busies itself with hypotheses, are two statements which the logician perceives come to the same thing. (CP 3.558, 1898)
In other words, because the method of mathematics is strictly deductive, its subject matter must be strictly hypothetical. As Peirce writes elsewhere ...
CSP: Mathematics is not subject to logic. Logic depends on mathematics. The recognition of mathematical necessity is performed in a perfectly satisfactory manner antecedent to any study of logic. Mathematical reasoning derives no warrant from logic. It needs no warrant. It is evident in itself. It does not relate to any matter of fact, but merely to whether one supposition excludes another. Since we ourselves create these suppositions, we are competent to answer them. But it is when we pass out of the realm of pure hypothesis into that of hard fact that logic is called for. We then find that certain modes of reasoning are sound, because they must, by mathematical necessity, be sound, in whatever universe there may be in which there is such a thing as experience. ...Mathematical reasoning holds. Why should it not? It relates only to the creations of the mind, concerning which there is no obstacle to our learning whatever is true of them. ... The only concern that logic has with this sort of reasoning is to describe it.
That being settled, I propose, by purely mathematical reasoning, to show that in any world in which there is such a thing as the course of experience--an element which is absent from the world of pure mathematics--in such world a certain kind of reasoning must be valid which is not valid in the world of pure mathematics. (CP 2.191-193, 1902)
Mathematical reasoning is always and only necessary reasoning about "creations of the mind," which the normative science of logic merely describes as deduction; but the hard facts of experience require probable reasoning, which the normative science of logic prescribes as induction. There is a mathematical/deductive aspect to the latter, to be sure--"All mathematical reasoning, even although it relates to probability, is of the nature of necessary reasoning" (CP 7.180, EP 2:82, 1901)--but this is true of all reasoning in every science.
Regards,
Jon Alan Schmidt - Olathe, Kansas, USA
Structural Engineer, Synechist Philosopher, Lutheran Christian
On Wed, Jul 14, 2021 at 11:29 AM Helmut Raulien <h.raul...@gmx.de> wrote:
Jon, ListI disagree with this "strictly hypothetical". In mathemathics, mostly there are hypotheses at the beginning, presumtions, which then are deductively proven or refuted by disproof or failed to prove. But, as I said, in mathematics also are surprising phenomena for subject matters (e.g. attractors in chaos theory), and in statistics the subject matter is induction. But the rules of investigation are always (strictly) deductive, I think, so to "its method is strictly deductive" I don´t have a counterexample. But to "its suubject matter is strictly hypothetical" I disagree.BestHelmut
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