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Sun 28 Feb, 2010 02:39 am

Why is pi irrational? I have a theory for that.....

Transcendental intuited space is*analog, or continuous.* Euclidean geometry is transcendentally intuited, or so it seems to me. We can conceive of perfectly straight lines, pristine equilateral triangles, ideal spheres.

But when we think of Euclidean or "transcendental" points, what is it we are thinking of? Is an ideal point infinitesimal? How thick is a plane? Is it infinitely thin?

Transcendental time is analog, or continuous. But humans cannot*speak* from transcendental (intuitive) time. Nor can humans do math *from *transcendental time. *Within it*, yes, but *from* it, no. Humans can only *think *of time* digitally*. Why? Because humans only *conceptualize* digitally.

Infinity is a useful but paradoxical negation of the finite. Why paradoxical? Because man's*concept* is *always* (transcendentally) *finite. *

*Science* *cannot* *perfectly measure anything.* Because science is not only number, but only *one* number. It's true that we use ten digits, but these are convenient shorthand. To think of two is to imagine a unity next to a unity. To think of a number like 66.7 is to think of a *unified quantity. *Quantity is a concept and a concept is *always* a unity, essentially a unity or essentially essence.

*does not compute. *This transcendentally spatial "ratio" is *analog*, and this relationship *cannot be quantized. *Pi is a collision of transcendental faculties. It's not only transcendental, but *doubly* trancendental, if you take my doubly transcendental meaning. For logos, as we use it every day, is also the collision of transcendentals (discrete and continous) //Edited for transcription error//author needs sleep, *bad...*

Transcendental intuited space is

But when we think of Euclidean or "transcendental" points, what is it we are thinking of? Is an ideal point infinitesimal? How thick is a plane? Is it infinitely thin?

Transcendental time is analog, or continuous. But humans cannot

Infinity is a useful but paradoxical negation of the finite. Why paradoxical? Because man's

Quote:

π(sometimes writtenpi) is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean space; this is the same value as the ratio of a circle's area to the square of its radius.

Quote:

It

π is an irrational number, which means that its value cannot be expressed exactly as a fractionm/n, wheremandnare integers. Consequently, its decimal representation never ends or repeats. It is also a transcendental number, which implies, among other things, that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) can be equal to its value; proving this was a late achievement in mathematical history and a significant result of 19th century German mathematics.

Twirlip

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Sun 28 Feb, 2010 03:33 am

@Reconstructo,

But Archimedes's favourite result, which at his request was engraved on his tombstone, was that the ratio of the volume of a sphere to the volume of its circumscribing cylinder is 2:3. Shouldn't that, by your reasoning, be an irrational number?Also (minor correction): pi is the ratio of circumference to diameter, not circumference to radius.

Reconstructo

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Sun 28 Feb, 2010 03:37 am

@Twirlip,

Twirlip;133494 wrote:

Also (minor correction): pi is the ratio of circumference to diameter, not circumference to radius.

Ah yes. I know. I've had 6 hours of sleep in 3 days. Thanks for point it out. I fixed it.

---------- Post added 02-28-2010 at 04:45 AM ----------

Twirlip;133494 wrote:

But Archimedes's favourite result, which at his request was engraved on his tombstone, was that the ratio of the volume of a sphere to the volume of its circumscribing cylinder is 2:3. Shouldn't that, by your reasoning, be an irrational number?

No, I wouldn't say so. Quite a few geometrical relationships can be quantized. Most of them perhaps. But it's the points at which they

I am grateful to learn about that ratio. I think it works because it's the circumscription of circumscription. I suspect that it's the straight line relating to the curve that will manifest the most analog-digital dissonance.

What can I say? Lately I only care about the transcendental. Math and geometry. Digital and analog. Time penetrating space. (Logos is a history with a future, attached to zeroes. Number is only eternity and it's negation, or -1.)

Quote:

Albert Einstein, on the other hand, stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."[6]

Twirlip

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Sun 28 Feb, 2010 04:13 am

@Reconstructo,

Quote:

Again, not necessarily. For example, the arc length of an arch of the cycloid generated by a rolling circle of radius I suspect that it's the straight line relating to the curve that will manifest the most analog-digital dissonance.

Cycloid - Wikipedia, the free encyclopedia

Reconstructo

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Sun 28 Feb, 2010 04:35 am

@Twirlip,

Twirlip;133504 wrote:

Again, not necessarily. For example, the arc length of an arch of the cycloid generated by a rolling circle of radiusris8r.

Cycloid - Wikipedia, the free encyclopedia

Well, I can't accept this as a valid example, but I'm not denying that you can find one. I don't think that my case rests on this anyway, but I'm open to being convinced. If you look at the cycloid it's still just curve on curve, despite being structured by a straight line.

Quote:

Acycloidis the curve defined by the path of a point on the edge of circular wheel as the wheel rolls along a straight line. It is an example of a roulette, a curve generated by a curve rolling on another curve.

One of my favorite philosophers described God as a polygon with an infinite number of sides..... Nicholas of Kues - Wikipedia, the free encyclopedia

Fil Albuquerque

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Sun 28 Feb, 2010 05:12 am

@Reconstructo,

I will just follow as my knowledge on maths is limited...(I use metaphor and words to replace maths purpose...!) But this is indeed very interesting...please continue and keep it "non-technical" but sustain detail...
Reconstructo

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Sun 28 Feb, 2010 05:31 am

@Fil Albuquerque,

Fil. Albuquerque;133518 wrote:

I will just follow as my knowledge on maths is limited...(I use metaphor and words to replace maths purpose...!) But this is indeed very interesting...please continue and keep it "non-technical" but sustain detail...

It all just hit me tonight. Between the conversation I've had on here with you and some similar conversations with a temporal-spatial friend, it's all coming together. Zeno applied digital logic, which is always and only the One is disguise, to intuitional continuous time and to intuitional continuous space. None of it computes. It's a transcendental collision. Do you know any Calculus? I don't know much, but enough to know that I'm about to know more. It addresses this "impossible"/transcendental/eternal collision. (For mortal man is the collision of 1 and 0, concept and continuity, number and qualia, the twin transcendentals, HCE and ALP..)

Calculus is Fancy Footwork, yes, but

Time is made of conceptualized qualia, which can

Parmenides - Wikipedia, the free encyclopedia

Zeno's paradoxes - Wikipedia, the free encyclopedia

Quote:

Calculus is usually developed by manipulating very small quantities. Historically, the first method of doing so was by infinitesimals. These are objects which can be treated like numbers but which are, in some sense, "infinitely small". An infinitesimal numberdxArchimedean property. From this point of view, calculus is a collection of techniques for manipulating infinitesimals.

Applications of differential calculus include computations involving velocity and acceleration, the slope of a curve, and optimization.

Twirlip

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Sun 28 Feb, 2010 10:41 am

@Reconstructo,

It is somewhat miraculous that mathematical analysis is able to treat a continuum as if it were a bunch of discrete points - and get away with it! But it took centuries of efforts by the brightest minds to get the calculus to work properly. Berkeley's More broadly, a maddening intuition about the discrete existing within the continuous has been tormenting me for more than thirty years, and I can never get it straight. For me, it is all bound up with equally confused intuitions about gender. (I'm transgendered, or gender-dysphoric, or something.) Masculinity has something to do with the discrete, and femininity with the continuous - of that, I'm

How's that for off-topic? :surprised::offtopic:

There's something rather male-centred about mathematical analysis, and set theory as a foundation for mathematics. Is that any more on-topic? :confused:

---------- Post added 02-28-2010 at 05:07 PM ----------

Quote:

I think I may have managed as much as 4 hours last night, although it took me about 14 hours in bed (on and off) to get that much. I don't feel right with less than about 8, although I can manage with 6, or at a pinch 5.I've had 6 hours of sleep in 3 days.

My sleep has been disturbed almost every night for three and a half years by two extremely aggressive and irrational neighbours who live directly above me. (A married couple, even older than myself.) I think they have both sworn at me on literally every occasion I have spoken to them, the man physically threatened me in 2007, and I haven't dared to speak to either of them since. Every effort to do anything about the problem rationally and practically has failed. I'm convinced I'm in some kind of Hell (and indeed that we all are). I wake up thinking of Hell. I retreat into total silence, because words fail me. Recently I've been having strange religious dreams.

So, if you're getting even

prothero

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Sun 28 Feb, 2010 01:34 pm

@Reconstructo,

I think the point partical theory of reality and matter is passe.Along with it the notion that space and time are continuous.

3 of the 4 fundamental forces are quantitized and discontinous.

It seems likely that space and time are also quantized and discontinous.

That fits with the process notion that reality is composed of discrete events "becoming" not "being" and that time is merely the change of process. Transcendental time is a mental construct not a reality of nature?

Twirlip

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Sun 28 Feb, 2010 02:06 pm

@prothero,

prothero;133605 wrote:

I think the point partical theory of reality and matter is passe.

It can't be dismissed as

prothero

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Sun 28 Feb, 2010 03:08 pm

@Twirlip,

Twirlip;133614 wrote:

You are right. General relativity (our current theory of gravity) describes point particles and a space time continuum. This is the reason that we have no unified field theory (grand unified theory, theory of everything) which combines the four fundamental forces. It can't be dismissed aspasseuntil somebody has reconciled quantum mechanics with general relativity, because the latter describes space-time as a 4-dimensional Lorentzian manifold, identifying point-events with quadruples of real numbers. (I don't know the exact technical description which would make this statement mathematically precise. Throw in the word "locally", or something! But the point - no pun intended - is that general relativity describes space-time as some kind of continuum, which is composed of point-events, which have to be described, in any given reference frame, by means of real numbers, which are in principle exact. I'm very much open to correction on the details. I never did get around to studying this stuff - which I regret.)

The other 3 forces (Electromagnetism, weak nuclear, strong nuclear) all have quantum descriptions and have been combined. It may turn out that a relativity like mathematical theory of the other forces emerges but it seems more likely that a quantum theory of gravity will emerge and allow the development of a GUT theory (string and now M-theory being the only strong candidates at the moment). Science is full of suprises and often leaps ahead due to some unforeseen spectacular breakthrough. Stay tuned.

I am just suggesting space and time are discrete, quantitized and discontinous for fun. Also it fits with process philosophy in which reality is composed of events not particles. It appeals for reasons of elegance, symmetry, simplicity and beauty. The correct answer often has these characteristics. I am no expert in these matters either.

Just as an aside quantum theories despite the difficulty in constructing conceptual models fit for our mental categories has been one of the most spectacularly successful and predictive theories in the history of science. General relativity gives infinite answers at the extremes of its equations. There is clearly something incomplete about general relativity.

Reconstructo

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Sun 28 Feb, 2010 07:54 pm

@prothero,

prothero;133605 wrote:

I think the point partical theory of reality and matter is passe.

Along with it the notion that space and time are continuous.

3 of the 4 fundamental forces are quantitized and discontinous.

It seems likely that space and time are also quantized and discontinous.

That fits with the process notion that reality is composed of discrete events "becoming" not "being" and that time is merely the change of process. Transcendental time is a mental construct not a reality of nature?

Yes indeed! Physics time is already and

It's important to distinguish, as perhaps you already have, that transcendental space is an intuitive space. The only

Space and time are only quantized/discontinuous to the degree that we impose number & word on them. It's not that space and time in themselves are one or the other, but that man is the collision of transcendentals. This is not a collision twixt man and nature, but a collision of human faculties.

There

---------- Post added 02-28-2010 at 09:04 PM ----------

Twirlip;133569 wrote:

More broadly, a maddening intuition about the discrete existing within the continuous has been tormenting me for more than thirty years, and I can never get it straight. For me, it is all bound up with equally confused intuitions about gender. (I'm transgendered, or gender-dysphoric, or something.) Masculinity has something to do with the discrete, and femininity with the continuous - of that, I'mfairlysure, but this is a mystical realm in which pretty much everything is paradoxical, and the opposite is almost as likely to be true as well. So, for me, the miracle of the infinitesimal calculus symbolises being both female and male at once - Freud's famous 'phallic woman', perhaps, or (a little more intelligibly to me) the baby as penis, the clitoris as penis, and Ferenczi's concept (inThalassa, which I didn't get at all far with reading) of the narcissistic basis of the Oedipus Complex.

This is not strange to me at all. Freud/Jung/Joyce. Joyce's Finnegans Wake might appeal to you on this. Man is a castle (built of digital bricks) beside a river. I just now understand

Yes, there is a numinous element to the 1 and 0. This Beauty is the Splendor of Truth. Blake knew it. Plato knew it. Joyce knew. The Phallus is the One is the Baby is the I. And also the negative sign that makes synthesis possible. The Triangle tells the story. Delta Theta, etc.

Reconstructo

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Mon 1 Mar, 2010 12:31 am

@Reconstructo,

Two kinds of transcendental processing applied to "one" pseudo-quantized flux, the source of which is noumena, unknown on principle. Time and space are transcendentally experienced as singular continuums, upon which we impose the digital transcendental of mathema and the synthetic transcendental of logos. Sound is experienced in intuitional time. Sight is experienced in transcendental space. Nose, tongue, touch: also in transcendental time. Logos evolves(and already

prothero

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Mon 1 Mar, 2010 09:44 pm

@Reconstructo,

[QUOTE=Reconstructo;133932]I completely agree. In my opinion, number and ideal geometry are transcendental, or eternal. As Schopenhauer said: "truth is fairest naked." The transcendental is also numinous, holy, sacred. Thus sacred geometry, and the music of the spheres. "Beauty is the splendor of truth." To gaze on the transcendental is to To tell you the truth I have trouble keeping up with you.

Compared to the pace of my ability to think and write you seem manic and "flight of ideas" although always interesting and full of educated references.

I notice that you are fixated a bit on numbers (digital) and reality (analog or continuous) or so you seem to claim, in a lot of recent posts on various topics.

I do not know, numbers and math, and the mathematical representation of reality is a profoundly important subject to ponder.

In my mind math and numbers are a form of logic, logic is a form of reason, and reason (logos) is a product of intelligence (nous). So the mathematical representation of reality is an indication that some form of intelligence (mind, not human consciousness or human mind) is behind the structure of the universe. Thus reason and hence math (numbers) are inherent in nature and an indication of rational intelligence as the basis for reality.

Reality itself I would maintain is also discrete, digital and quantitized and the impression of continuity in space and time is a product of the conceptual constructs of the human mind. (Much as we can digitize any medium and experience it as continuous). Continuity is a feature of human conception not independent reality. This would make number even more foundational to reality.

Ours ideas of such continuous perfect forms as circles, spheres, etc. can only be represented in mathematical form not in reality. The calculus the discrete representation of the continuous is thus a major breakthrough in the ability to provide a mathematical model of reality.

The only one of the four fundamental forces which is mathematically continuous and point particle is (gravity, general relativity) and the equations of general relativity give infinite and other nonsense results at the extremes, so that theory is generally considered incomplete and an approximation to a complete theory. Gravity will probably end up in a quantum formulation as well which would effectively digitize or quantitize space- time as well.

All this fits perfectly with the notion of process philosophy which provides that reality is composed of discrete events (moments of experience) and not of inert, insensate "material" point particles situated in continuous space time.

Well anyway just a sketch of number, reason, quanta, and process and the ways in which they correlate. Is number then transcendent or transcendental? I am not sure how you use the term but in theology transcendental means beyond the material realm, or separate from the material realm and so number I would maintain along with reason exists prior to "creation". Number is a form of truth, of beauty, and a transcendental reality not at all a construction of the human mind imposed on reality but Truth discovered not invented.

Reconstructo

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Mon 1 Mar, 2010 10:52 pm

@prothero,

prothero;134350 wrote:

In my mind math and numbers are a form of logic, logic is a form of reason, and reason (logos) is a product of intelligence (nous).

I salute your engagement of this issue.

In my opinion, Logos is not

In fact, I suggest that number is created by subtracting or negating this analog element of logos. Hegel goes into this. Also, I attack this in my "Truth is Triangular" thread. Logos is synthetic, or temporal. Number is equation or eternity.

---------- Post added 03-01-2010 at 11:56 PM ----------

prothero;134350 wrote:

Reality itself I would maintain is also discrete, digital and quantitized and the impression of continuity in space and time is a product of the conceptual constructs of the human mind. (Much as we can digitize any medium and experience it as continuous). Continuity is a feature of human conception not independent reality. This would make number even more foundational to reality.

I have to disagree w/ you here. Reality is experience both digitally and continuously. Do you know Zeno's paradoxes? The thing is, we

I do agree that continuity is a transcendental intuition. But I also think that

---------- Post added 03-02-2010 at 12:01 AM ----------

prothero;134350 wrote:

Ours ideas of such continuous perfect forms as circles, spheres, etc. can only be represented in mathematical form not in reality. The calculus the discrete representation of the continuous is thus a major breakthrough in the ability to provide a mathematical model of reality.

I think we must distinguish between perfect geometrical forms and numbers. Perfect geometry, in my view, is possible only

I don't think that the continuous

---------- Post added 03-02-2010 at 12:05 AM ----------

prothero;134350 wrote:

Well anyway just a sketch of number, reason, quanta, and process and the ways in which they correlate. Is number then transcendent or transcendental? I am not sure how you use the term but in theology transcendental means beyond the material realm, or separate from the material realm and so number I would maintain along with reason exists prior to "creation". Number is a form of truth, of beauty, and a transcendental reality not at all a construction of the human mind imposed on reality but Truth discovered not invented.

I'm coming from a Kantian use of the word "transcendental." For me, the the "transcendent" is just an adjective for the

I differ w/ you in thinking that number is possibility just an imposition of the human mind. But I can't be sure.

Reconstructo

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Fri 5 Mar, 2010 07:08 pm

@Reconstructo,

Time is experienced spatially, one might think, considering music. But we only think of it numerically. Or do we?
Zetetic11235

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Thu 11 Mar, 2010 02:35 pm

@Reconstructo,

I have considered this very explicitly in the past. I think that many of these concepts have to do with a grammatical recursive-ness built into certain verbs and nouns, and in fact these point to very specific concepts with recursion built into them.

If we consider the intuition of a 'perfect' sphere, we are actually speaking of an initial state object

If we look at the density of Q^3 and the density of R^3, the real 3-space in calculus and analytic geometry, we notice something astonishing from the chosen indexing schemes: If we take every point from Q^3 out of R^3 we still have a space that is essentially identical to R^3.

To represent the transcendental nature of geometry, all that is needed is infinite density, and Q^3 provides that. R^3 is a completion of Q^3, it allows for it to be the case that (this is the Cauchy, rather than Dedekind perspective) any infinite sequence of numbers in Q^3 has a limit point (so for instance PI is in this because it is the only limit point for the sequence {3, 3.1, 3.14, 3.141, 3.1415...} each term of which can be discovered by using the infinite series for PI and truncating it to a finite sum to avoid taking a limit).

This is a very transcendental (if I am thinking of the word from the right perspective) place to approach it from, the concept of a limit being quite odd and transcendental itself. In this way I concur with your assertion that numbers such as PI are dually transcendental, and this is due to the singly transcendental nature of a perfect circle, since it requires infinite density. So when we describe the circumference of a circle divided by it's diameter we implicitly require an infinitely recursive and thus transcendental scheme, and furthermore we are speaking of a

In realizing that we can derive these transcendental numbers from examining the relationship between singly transcendental concepts, we begin looking at appending these new numbers, but also we ask if these 'numbers' are really numbers at all! They are in principle and origin and behavior so very different and contain so much more conceptual baggage than other numbers! I think that Wittgenstein would say that calling these 'numbers' numbers is the result of confusion!

Reconstructo

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Fri 12 Mar, 2010 12:18 am

@Zetetic11235,

Zetetic11235;138690 wrote:

This is a very transcendental (if I am thinking of the word from the right perspective) place to approach it from, the concept of a limit being quite odd and transcendental itself. In this way I concur with your assertion that numbers such as PI are dually transcendental, and this is due to the singly transcendental nature of a perfect circle, since it requires infinite density. So when we describe the circumference of a circle divided by it's diameter we implicitly require an infinitely recursive and thus transcendental scheme, and furthermore we are speaking of atwo transcendental schema! So we have reached a second tier of transcendence (I do kind of dislike that term though).relationship between

In realizing that we can derive these transcendental numbers from examining the relationship between singly transcendental concepts, we begin looking at appending these new numbers, but also we ask if these 'numbers' are really numbers at all! They are in principle and origin and behavior so very different and contain so much more conceptual baggage than other numbers! I think that Wittgenstein would say that calling these 'numbers' numbers is the result of confusion!

I enjoyed your post. It's a fascinating subject. For me, one of the transcendentals is number itself, including integers. The other would be the sort of infinite density you mention. A number line is like a Euclidean line. We get an infinite density of numbers/cuts, as many as we want between

As far as logos goes, or usual human discourse, this gets more complicated as words often refer to organized qualia. Still, on a logical level I think the same reduction holds.

Zetetic11235

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Fri 12 Mar, 2010 03:42 pm

@Reconstructo,

I gather that you are using AND and NOT figuratively? If not, in the literal first order logic meaning they are sufficient. Or can be derived in this way ~(~P^~P). So then we also get implication with P V ~Q. How about neither-nor? Sheffer stroke - Wikipedia, the free encyclopediaYou should read Carnap's Logical Syntax. I unfortunately haven't read it yet, but I have read Quine's analysis of it and it seems very interesting. From what I understand he tries to flesh out and formalize Wittgenstein's picture theory of meaning. That is the most comprehensive reductionist programme I know of when it comes to language. Quine has some very interesting things to say about Carnap's ideas and vice versa (read 'Dear Carnap, Dear Van-the Quine Carnap Correspondence', it's interesting, espceially the lectures at the beginning).

Reconstructo

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Sat 20 Mar, 2010 02:25 pm

@Zetetic11235,

Zetetic11235;139131 wrote:

I gather that you are using AND and NOT figuratively? If not, in the literal first order logic meaning they are sufficient. Or can be derived in this way ~(~P^~P). So then we also get implication with P V ~Q. How about neither-nor? Sheffer stroke - Wikipedia, the free encyclopedia

You should read Carnap's Logical Syntax. I unfortunately haven't read it yet, but I have read Quine's analysis of it and it seems very interesting. From what I understand he tries to flesh out and formalize Wittgenstein's picture theory of meaning. That is the most comprehensive reductionist programme I know of when it comes to language. Quine has some very interesting things to say about Carnap's ideas and vice versa (read 'Dear Carnap, Dear Van-the Quine Carnap Correspondence', it's interesting, espceially the lectures at the beginning).

Yes, NAND is great! I'm no expert, but I actually bumped into a formal logic textbook about 20 years ago at Waldenbooks, and I was taken with its beauty. I loved that 3 stroke equals sign, logic trees, etc. Yes, I am using "and" and "not" figuratively, as the words involved are contingent. Let us say tautology and contradiction, as Wittgenstein does. Posit or negate. And negation is a positing, the positing of a determinate negation. To posit "not p" is significant.

They key for me is that we only think in singularities. That's a bit of an overstatement to get the point across. We can't think the infinite or unbounded, because thinking

Do you agree that formal logic is transcendental? A = A. How does one prove that? And of course we don't bother, as thing is obvious. Fichte was on to this. He tried also to root out the structure of the mind.

For me the picture theory of meaning is Wittgenstein showing that all the conversation that really matters to us is trans-logical or sub-logical. It's outside the bounds of pure tautology and contradiction. Our words are stained with qualia and feeling, else they would be as useless as tautologies. Forgive me if I ramble. The coffee is strong this morning.

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