On 3/17/2013 3:48 PM, Virgil wrote: > In article > <firstname.lastname@example.org>, > WM <email@example.com> wrote: > >> On 17 Mrz., 08:18, fom <fomJ...@nyms.net> wrote: >>> On 3/16/2013 4:37 PM, WM wrote: >>> >>>> On 16 Mrz., 21:19, Virgil <vir...@ligriv.com> wrote: >>> >>>>>> In potential infinity there is no necessary line except the last one. >>>>>> We know that with certainty from induction. Every found and fixed line >>>>>> n cannot be necessary, because the next line contains it. >>> >>>>> AS soon as something is identifies as a natural or a FIS of the set of >>>>> naturals, it has a successor. It cannot be either a natural nor a FIS of >>>>> the naturals without a successor. at least by any standard definition of >>>>> naturals. >>> >>>> As soon as a second becomes presence, it has a successor. >>> >>> And what fantasy is this? >>> >>> The successor to the present has existential form but >>> has not yet happened. >>> >>> That is not the Kantian aprioriticity of time. >>> >>> That is not the Hegelian becoming of the present. >>> >>> It is the unfounded object of unjustifiable belief. >> >> It is the well known and established natural way how time passes and >> how the system of human actions in time goes off. > > Mathematical truth is independent of time.
Well that depends on how a philosophy which makes such a statement addresses the issue.
Frege specifically addresses the sense expressed by WM:
"Next there may be those who will prefer some other definition as being more natural, as for example the following:
if starting from x we transfer our attention continually from one object to another to which it stands in relation phi, and if by this procedure we can finally reach y, then we say that y follows in the phi-series after x.
"Now this describes a way of discovering that y follows, it does not define what is meant by y's following. Whether as our attention shifts, we reach y may depend on all sorts of subjective contributory factors, for example, on the amount of time at our disposal or on the extent of our familiarity with the things concerned. Whether y follows in the phi-series of x has nothing to do with our attention and the circumstances in which we transfer it; on the contrary, it is a question of fact, just as much as it is a fact that a green leaf reflects light rays of certain wavelengths, whether or not these fall into my eye and give rise to a sensation, and a fact that a grain of salt is soluble in water whether or not I drop it into water and observe the result, and a further fact that it remains still soluble even when it is utterly impossible for me to make any experiment with it.
"My definition lifts the matter into a new plane; it is no longer a question of what is subjectively possible but of what is objectively definite. For in literal fact, that one proposition follows from certain others is something objective, something independent of the laws that govern the movements of our attention, something to which it is immaterial whether we actually draw the conclusion or not. What I have provided is a criterion which decides in every case the question "Does it follow after?" wherever it can be put; and however much in particular cases we may prevented by extraneous difficulties from actually reaching a decision, that is irrelevant to the fact itself.
Although Frege eventually retracted his own definition, what he is saying here is that defined relations relative to a defined logic constitute the matter of an objective mathematics.
To understand the distinction, one may contrast Frege with Weyl. The latter is, at least, tentatively willing to admit a set theoretic ground that does not yield the transfinite. He says this in reference to Dedekind:
"A set-theoretic treatment of the natural numbers such as that offered by Dedekind may indeed contribute to the systematization of mathematics; but it must not be allowed to obscure the fact that our grasp of the basic concepts of set theory depends on a prior intuition of iteration and of the sequence of natural numbers."
Let me give credit to WM for rejecting Dedekind. He has shown enough consistency to realize that a Dedekindian ground is a ground that fixes the successor relation with respect to a system that constitutes a completed infinity. Weyl apparently misses the inconsistency of his position out of desire to reject transfinite arithmetic.
But, a few pages earlier, Weyl makes an interesting statement concerning the nature of "objective" fact.
Note the explicit rejection of logic and definition in his statement,
"Therefore, how two sets (in contrast to properties) are defined (on the basis of the primitive properties and relations and individual objects exhibited by means of the principles of section 2) does not determine their identity. Rather, an objective fact which is not decipherable from the definition in a purely logical way is decisive; namely, whether each element of the one set is an element of the other, and conversely. [...]"
So, as a reader of this statement, I am first expected to reject prior definitions and to reject logical relations. Then, I am expected to understand the discursive assertion explaining what it is that cannot be explained.
However, I am to understand that this is sensible with respect to some other prior principles explained elsewhere. And, I am to understand that what cannot be explained to me can sensibly be expressed as a rule.
The statement goes on to say,
"Moreover, we see that the description of a finite set in individual terms is, considered formally, just a special case of that based on a rule. For example, if a,b,c are three objects of our category, then
is the judgement scheme of the derived property 'being a or b or c'; and the set having just those three objects as its elements correspond to this property."
What is relevant from section 2 that I am expected to not ignore while being told to ignore is the following:
"By simple or primitive judgment scheme we mean those which correspond to the individual immediately given properties and relations. To these we add the identity scheme J(xy) (meaning 'x is identical to y' i.e., 'x=y')"
So, once again, the situation resolves to the concept of "immediately given individual properties" or the objective fact that the purport of singular reference suffices as an establishment of singular reference.
And, once again, searching through these philosophies and the definitions leads to the fact that presentations of Leibniz law such as
misrepresents what, in fact, Leibniz actually wrote:
"What St. Thomas affirms on this point about angels or intelligences ('that here every individual is a lowest species') is true of all substances, provided one takes the specific difference in the way that geometers take it with regard to their figures."
Returning to how time might inform mathematics in relation to arithmetical progressions, there is Kant:
1) Time is not an empirical concept that is derived from experience. [...]
2) Time is a necessary representation that underlies all intuitions [...]
3) The possibility of apodeictic principles concerning the relations of time, or of axioms of time in general is grounded upon this a priori necessity. [...] We should only be able to say that common experience teaches that this is so; not that it must be so. These principles are valid as rules under which alone experiences are possible; and they instruct us in regard to experiences, not be means of them.
4) Time is not a discursive, or what is called a general concept, but a form of pure sensible intuition.
5) The infinitude of time signifies nothing more than that every determinate magnitude of time is possible only through limitations of one single time that underlies it."
And, should anyone who wishes to reject the mathematical aspect of Kant's philosophy in relation to his remarks here on the basis of nineteenth and twentieth century "progress" one need only consider the author to whom George Greene directed me to learn about why Kant had become outdated. Boolos writes:
"[Crispin] Wright regard's Hume's principle as a statement whose role is to fix the character of a certain concept. We need not read any contemporary theories of the a priori into the debate between Frege and Kant. But Frege can be thought to have carried the day against Kant only if it has been shown that Hume's principle is analytic, or a truth of logic. This has not been done. [...]
"Well. Neither Frege nor Dedekind showed arithmetic to be a part of logic. Nor did Russell. Nor did Zermelo or von Neumann. Nor did the author of Tractatus 6.02 or his follower Church. They merely shed light on it."
And, George's recommendation seemed particularly odd when I found that Boolos quoted Hao Wang's remark,
"The reduction, however, cuts both ways. It is not easy to see how Frege can avoid the seeming frivolous argument that if his reduction is successful, one who believes firmly in the synthetic character of arithmetic can conclude that Frege's logic is thus proved to be synthetic rather than that arithmetic is proved to be analytic."
So, one may clearly hold the contrary to Virgil's statement depending on how one views time and its relationship to mathematical thought without, apparently, being in too much mathematical jeopardy.