Date: Mar 17, 2013 7:17 PM Author: fom Subject: Re: Matheology § 224 On 3/17/2013 3:48 PM, Virgil wrote:

> In article

> <ded07b82-5853-4a1e-af1d-a68c3ef695aa@j9g2000vbz.googlegroups.com>,

> WM <mueckenh@rz.fh-augsburg.de> 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

P(x)=J(xa)+J(xb)+J(xc)

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

http://plato.stanford.edu/entries/identity-relative/#1

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."

Leibniz

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.