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AppendixEdit
This appendix is for Reading Philosophical Investigations: Article Content.
Natural life and infinityEdit
The restriction to potential infinities, which I consider only logical, is tied up with the "this life only" viewpoint. The concept of actual infinity is part of the usual concept of God---an actual infinity of actual infinities. God can never be thought of as a potential infinity because it would mean God can never be complete, a very untheological idea. This would also mean that the idea of an eternal afterlife would be out of the question.
Concepts of the nature of God are "bedrock" for many people. My main point is that arguing to prove my assumptions are true would be pointless: if not in theory at least in historical reality. Even in theory, Wittgenstein in On Certainty^{[1]} saw a person having a right to hold to a founational belief if abandoning it would leave him unable to form any judgements at all. Calling it "knowledge" would be wrong however.
I think it would be unwise to articulate further on these two assumptions at this point. The explanation of the two concepts of infinity will definitely by described at the proper time.
Return to reference #4 of Reading Philosophical Investigations: Article Content.
Examples to illustrate logical formsEdit
An example of logical form is provided by the sentences:
i) Every philosopher reads some book by Wittgenstein.
and
ii) Some philosopher reads every book by Wittgenstein.
The two sentences (i and ii, above) differ in logical form, because they differ in the placement of "every" and "some".
The sentences:
iii) Every philosopher reads some book by Wittgenstein.
and
iv) Every politician reads some book by Churchill.
have the same logical form.
The English sentence:
v) No lawyer who saw every patient respects some doctor.
can be rendered in formal logical form using the notation of Gottlob Frege:
¬∃x{L(x) & ∀y[P(y) → S(x,y)] & ∃z[D(z) & R(x,z)]} (see)
Return to reference #6 of Reading Philosophical Investigations: Article Content.
Principia MathematicaEdit
"Principia Mathematica was really much more about language than about mathematics. It was an attempt to find a second order or derived language in symbolism that would do away with the ambiguities and vagaries of natural languages -language that seemed to hide and confuse the expressions of philosophy and logic." This statement from the main text of the article can be contrasted with what is said at wikipedia:
"It is an attempt to derive all mathematical truths from a well-defined set of axioms and inference rules in symbolic logic."
and at the Stanford Encyclopedia of Philosophy:
"Written as a defense of logicism (i.e., the view that mathematics is in some significant sense reducible to logic) the book was instrumental in developing and popularizing modern mathematical logic. It also served as a major impetus for research in the foundations of mathematics throughout the twentieth century. Next to Aristotle's Organon, it remains the most influential book on logic ever written."
From the perspective of some mathematicians, Principia Mathematica showed that much of mathematics can be formalized starting from a foundation in logic (if you assume that Cantor's set theory is valid). If much of mathematics can be formalized, how much of human thought, reasoning and philosophy can be formalized?
My description of the goal of Principia Mathematica is not really in conflic with the two quotes from other sources (above). We all agree on what was attempted. If there is a contrast it is in the conclusion of Wittgenstein that Symbolic Logic was a failure. The Analytical traditon that began with Frege and Russell AND Wittgenstein in the Tractatus has in practice abandoned the premisses of the founders but keep the Platonism of Frege and the Scientism of Russell. Even in the Tractatus, Wittgenstein did not equate logic and mathematics, nor did he care for Russell's aim to make philosophy scientific. The conclusion of the Tractatus was that having achieved his goal, little was accomplished. My treatment of the Tractatus explains what I believe Wittgenstein accomplished by the Tractatus.
Wittgenstein's mathematics is certainly not mainstream in its philosophical leanings, but he had no argument with mathematicians as long as the were doing mathematics. However he did think "set theory" was a cancerous growth on mathematics. This is the area of his rejection of Cantor and Transfinite Arithmetic, and the focus on infinity. This plays a key role in the part of Philosophical Investigations on following rules, which is discussed in a later article in the series "Aid and Commentary on Ludwig Wittgenstein's PHILOSOPHICAL INVESTIGATIONS".
Return to reference #7 of Reading Philosophical Investigations: Article Content.
Analytic AssumptionEdit
As discussed by Paul Newall, a key assumption tested by analytical philosophy is often called the "logistic thesis" and is frequently traced back to Russell and Whitehead and their efforts in Principia Mathematica: can mathematics be reduced to logic? Assuming that Principia Mathematica had proved the logical foundation of mathematics, a key issue confronted by analytic philosophers was: "Can natural language be reduced to logic?"
Another assumption that seems to have been explored by Wittgenstein has been called the "Methodological Analytic Assumption" by Lynn Holt (personal communication)^{[2]}. The Methodological Analytic Assumption suggests that by analyzing the constituent parts of a system in isolation you can come to discover the relational properties which make the whole more than a simple "sum" of the parts. The idea that had to be explored was if there were "logical atoms" that could be combined by a set of "logical rules" to produce the sense of statements in natural language. If so, then the contents of human thought could be linked to language according to rules of logic within a parallel mental universe corresponding to the physical universe where physical atoms obey the laws of nature.
Within the system developed by Wittgenstein in the Tractatus, it was important that each atomic proposition could be true or false independent of any other atomic proposition. Wittgenstein's failure to find a way to naturally achieve this requirement played a role in his eventual abandonment of his theory of linguistic logical atomism (See the Stanford Encylcopedia of Philosophy's article on logical atomism).
Return to reference #9 of Reading Philosophical Investigations: Article Content.
Logic and EthicsEdit
From Wittgenstein's "Lecture on Ethics":
"Suppose one of you were an omniscient person and therefore knew all the movements of all the bodies in the world dead or alive and that he also knew all the states of mind of all human beings that ever lived, and suppose this man wrote all he knew in a bigbook, then this book would contain the whole description of the world; and what I want to say is, that this book would contain nothing that we would call an ethical judgment or anything that would logically imply such a judgment."
.....
"I said that so far as facts and propositions are concerned there is only relative value and relative good, right, etc. And let me, before I go on, illustrate this by a rather obvious example. The right road is the road which leads to an arbitrarily predetermined end and it is quite clear to us all that there is no sense in talking about the right road apart from such a predetermined goal. Now let us see what we could possibly mean by the expression, 'the absolutely right road.' I think it would be the road which everybody on seeing it would, with logical necessity, have to go,or be ashamed for not going.
And similarly the absolute good, if it is a describable state of affairs, would be one which everybody, independent of his tastes and inclinations, would necessarily bring about or feel guilty for not bringing about. And I want to say that such a state of affairs is a chimera. No state of affairs has, in itself, what I would like to call the coercive power of an absolute judge."
.....
"Now the answer to all this will seem perfectly clear to many of you. You will say: Well, if certain experiences constantly tempt us to attribute a quality to them which we call absolute or ethical value and importance, this simply shows that by these words we don't mean nonsense, that after all what we mean by saying that an experience has absolute value is just a fact like other facts and that all it comes to is that we have not yet succeeded in finding the correct logical analysis of what we mean by our ethical and religious expressions. Now when this is urged against me I at once see clearly, as it were in a flash of light, not only that no description that I can think of would do to describe what I mean by absolute value, but that I would reject every significant description that anybody could possibly suggest, ab initio, on the ground of its significance.
That is to say: I see now that these nonsensical expressions were not nonsensical because I had not yet found the correct expressions, but that their nonsensicality was their very essence. For all I wanted to do with them was just to go beyond the world and that is to say beyond significant language. My whole tendency and, I believe, the tendency of all men who ever tried to write or talk Ethics or Religion was to run against the boundaries of language."
Return to reference #15 of Reading Philosophical Investigations: Article Content.
ReferencesEdit
- ^ On Certainty by Ludwig Wittgenstein. Publisher: Harper Perennial (1972) ISBN: 0061316865.
- ^ The idea of a "methodological assumption" is implicit in: "Avoiding the Analytic Assumption in Theories of Practical Rationality: A Metatheoretical Note." Contemporary Philosophy, volume 13, issue 7, 1991.
Other wiki pages for this articleEdit
- Before reading the Philosophical Investigations: a Necessary Context
- Reading Philosophical Investigations: Article Content