##### Posted by: Holden

«**on:**May 21, 2021, 03:50:51 am »

"Where there is matter, there is geometry." ~ Johannes Kepler

... oscillates between clownery and anger, between mad laughter and sobbing.

FreeBoards.org Installed

"Where there is matter, there is geometry." ~ Johannes Kepler

Well, let me put this video in its proper context. Perhaps I should have done that when I posted the video in the first place.

This video went viral on the internet a few months back:

https://theconversation.com/is-mathematics-real-a-viral-tiktok-video-raises-a-legitimate-question-with-exciting-answers-145244

https://www.wsj.com/articles/a-viral-video-asks-a-deep-question-11599757497

The thing is,in the grade school they do not teach any math history or math philosophy at all.At least mine did not. I have checked the undergraduate math syllabus and I find that the math majors do get to learn a bit of math history and philosophy but by then many, many people end up quitting maths for good.

Now, I have read a bit of maths history and philosophy on my own but I have also learnt a point which you have made a number of times-**just philosophy of mathematics is not sufficient to comprehend mathematics properly**. To that end, I am working on mathematics fundamentals,putting in many ,many hours every week and even then it will be only be after quite a few years(if I managed to continue to work consistently), that I would be in a position to even start a genuine dialogue about mathematics with you.

But I wish that I could at least make that happen before I am pushing up the daisies.

This video went viral on the internet a few months back:

https://theconversation.com/is-mathematics-real-a-viral-tiktok-video-raises-a-legitimate-question-with-exciting-answers-145244

https://www.wsj.com/articles/a-viral-video-asks-a-deep-question-11599757497

The thing is,in the grade school they do not teach any math history or math philosophy at all.At least mine did not. I have checked the undergraduate math syllabus and I find that the math majors do get to learn a bit of math history and philosophy but by then many, many people end up quitting maths for good.

Now, I have read a bit of maths history and philosophy on my own but I have also learnt a point which you have made a number of times-

But I wish that I could at least make that happen before I am pushing up the daisies.

First reaction: (joking) "That girl needs a spanking."

After coffee settles in my system: "A girl like that might enjoy undermining a young man's confidence in his own mathematical maturity. Maybe the ancients were just being prudent or even wise when they isolated males for a mathematical education. If I was a young man studying, interactions with such a cutie would most likely leave a deep and long-lasting depressive impression on my psyche and its fragile faith in alphanumeric constructs. I would not be able to refute her."

Take 3: "She is a very*frustrated* one, isn't she? She is what one might call a hard nut to crack. Anything I say about this is not going to sound good, is it?"

After coffee settles in my system: "A girl like that might enjoy undermining a young man's confidence in his own mathematical maturity. Maybe the ancients were just being prudent or even wise when they isolated males for a mathematical education. If I was a young man studying, interactions with such a cutie would most likely leave a deep and long-lasting depressive impression on my psyche and its fragile faith in alphanumeric constructs. I would not be able to refute her."

Take 3: "She is a very

Quote from: Holden

I can see now why you would find Badiou interesting. He seems to be using the insights gathered from set theory in his philosophy.

Yes, interesting is the right word. I do not particularly "like" the man himself, not since seeing him and hearing him in the video you linked to in this thread. He dismissed pessimism so swiftly and arrogantly that I am afraid a great deal of deep-buried cultural animosities rose to the surface of my breast.

I apologize for being such a contradiction. I have read Badiou, and I am very interested in his set-theoretic approach, but he and I are totally different species of philosophers, and, if Badiou is a representative of a "mathematically grounded philosophy," then I would prefer going down in history with the mad angry poets.

Quote from: Holden

About Badiou : He pretends as if he has a direct line to “God”.

He is an optimist, enough said. A mind which is an optimist can never come up with the true mathematical philosophy.

Yes, Holden. Thank you for pointing these things out to me. I suggest we embrace our status as rejects, outcasts, and non-academic scholars.

It is only fitting that such Schopenhauer Disciples would represent the non-academic.

Maybe we can simply embrace a call to crank up the invocation of existential crises. Isn't that what Kant did that Schopenhauer found so appealing?

I am reading through this paper on neologicism left by Holden.

Of interest:

I'm just letting Holden know that I have eventually gotten to this link you left back in May. I am reading it slowly, that is, a little at a time, and continue to peck away in bursts and shots throughout today, in between the other "tasks," as long as there are no surprises waiting to shock me today. One never knows what Fate has in store for us. The uncertainty of things outside one's conscious control (including our unrequested needs and unrequested impulses) would justify at least a minimal amount of paranoia and fear or dread and anxiety.

Good times.

I read up into about page 14 but could tell my heart was not in it 100% ... I began to lose interest. I began to ask, why bother? I began to get depressed even.

Now, more than ever, I just want to be honest.

Of interest:

Quote

. . . , the following conclusion has been obtained, which is also known as the thesis of logicism concerning the nature of mathematics: Mathematics is a branch of logic. It can be derived from logic in the following sense:

a. All the concepts of mathematics, i.e., of arithmetic, algebra,and analysis, can be defined in terms of four concepts of pure logic.

b. All the theorems of mathematics can be deduced from those definitions by means of the principles of logic (including the axioms of infinity and choice).

As recently as 1945 ... some philosophers took set theory to be a part oflogic and thought logicism had been vindicated.Why does this sound so alien to our ears? In part under Quine’s influence, a picture emerged which made it natural to think that set theoryis not a part of logic.

I'm just letting Holden know that I have eventually gotten to this link you left back in May. I am reading it slowly, that is, a little at a time, and continue to peck away in bursts and shots throughout today, in between the other "tasks," as long as there are no surprises waiting to shock me today. One never knows what Fate has in store for us. The uncertainty of things outside one's conscious control (including our unrequested needs and unrequested impulses) would justify at least a minimal amount of paranoia and fear or dread and anxiety.

Good times.

Quote

Whitehead and Russell directly postulate the individuals needed for the construction of the natural numbers with the notorious axiom of infinity. Most philosophers have concluded that the axiom of infinity is obviously not a principle of logic, given its strong existential commitments. However, is it so obvious?

I read up into about page 14 but could tell my heart was not in it 100% ... I began to lose interest. I began to ask, why bother? I began to get depressed even.

Now, more than ever, I just want to be honest.

"It seems that there was a mathematician who had become a novelist. 'Why did he do that?' people in Gottingen marvelled. 'How can a man who was a mathematician write novels?' 'But that is completely simple,' Hilbert said. 'He did not have enough imagination for mathematics, but he had enough for novels.' "From Hilbert by Constance Reid.

Wir mussen wissen. Wir werden wissen. -Hilbert

(We must know.We will know)

Wir mussen wissen. Wir werden wissen. -Hilbert

(We must know.We will know)

I can see now why you would find Badiou interesting.He seems to be using the insights gathered from set theory in his philosophy.He says there is no transcendent unity.The unit as one operation brings together different things and forms sets.Before the sets are ever made, all there is, are multiplicities.

He wants to know how new possibilities might emerge in this process of counting and recounting,.There are always uncounted multiplicities bidding their time..

He says that truth is not “a relation of appropriateness between the intellect and the thing intellected, a relation of adequation which always supposes… that the truth be localizable in the form of a proposition.” Instead, “a truth is, first of all, something new.”

“For the process of a truth to begin, something must happen. What there already is, the situation of knowledge as such, only gives us repetition. For a truth to affirm its newness, there must be a supplement; this supplement is committed to chance. It is unpredictable, incalculable; it is beyond what it is. I call it an event. A truth appears in its newness because an eventful supplement interrupts repetition. Examples: the appearance with Aeschylus of theatrical tragedy, the irruption with Galileo of mathematical physics, an amorous encounter which changes a whole life, or the French Revolution of 1792”

He wants to know how new possibilities might emerge in this process of counting and recounting,.There are always uncounted multiplicities bidding their time..

He says that truth is not “a relation of appropriateness between the intellect and the thing intellected, a relation of adequation which always supposes… that the truth be localizable in the form of a proposition.” Instead, “a truth is, first of all, something new.”

“For the process of a truth to begin, something must happen. What there already is, the situation of knowledge as such, only gives us repetition. For a truth to affirm its newness, there must be a supplement; this supplement is committed to chance. It is unpredictable, incalculable; it is beyond what it is. I call it an event. A truth appears in its newness because an eventful supplement interrupts repetition. Examples: the appearance with Aeschylus of theatrical tragedy, the irruption with Galileo of mathematical physics, an amorous encounter which changes a whole life, or the French Revolution of 1792”

For Herr Hauser:

https://mally.stanford.edu/Papers/neologicism2.pdf

https://mally.stanford.edu/Papers/neologicism2.pdf

I have said it before and I will say it again -your profound interest is mathematics is getting rubbed off on me.

Hopefully,in the near future I will be able to share some

things with you(mathematical) that you would find interesting too.

Hopefully,in the near future I will be able to share some

things with you(mathematical) that you would find interesting too.

I had to highlight 2nd Saturdays and 4th Saturdays in the months. I found that the following formula does the job:

=IF(N7=FLOOR(DATE(YEAR(N7),MONTH(N7),14),7),"Second

Saturday",IF(N7=FLOOR(DATE(YEAR(N7),MONTH(N7),28),7),"Fourth Saturday",""))

=IF(N7=FLOOR(DATE(YEAR(N7),MONTH(N7),14),7),"Second

Saturday",IF(N7=FLOOR(DATE(YEAR(N7),MONTH(N7),28),7),"Fourth Saturday",""))

I may have experienced a breakthrough in incorporating Schopenhauer's "artistic" influence into my "mathematical understanding."

I have always saught, as a kind of Holy Grail, some unattainable Pure Intuitive Quasi-Instinctual Understanding of Mathematical Phenomena, and I might have fantasized that Schopenhauer possessed such an understanding. He may have found the symbolic coneptualized [FORMULATED] representation grotesque and technical, perhaps in a way analogous to ancient Hebrews finding any attempt to pronounce the Tetragrammaton YHWH offensive.

I'm not sure, but the Schopenhauer connection appears to be all about UNDERSTANDING. I have always romanticized what it might feel like to expereince a deepening of genuine understanding, rather than be mired down in the frustratingly tedius details of proof and code.

As I have aged, with my memory becoming even less sharp, I have had to fall back on VISUALIZATION in order to translate "words" into numerical representation.

This is mathematical thinking from the purely Schopenhauerian perspective, that is, in the privacy of the environment between our ears in honest contemplation and reflective consciousness, do we understand?

A textbook example where I apply Schopenhauerian Visualization would be appropriate. Holden suggested that Schopenhauer might hold the key to help the flies find our way out of the jar. I think that a key to this puzzle can be discovered in "word problems" --- where one must translate a verbally stated problem into some formula. That is the challenging part. Once that is formula, one might apply the necessary operations, such as discovering when the derivative is zero in Maximize/Minimize problems.

_______________________

Consider:

A chatering company will provide a plane for a fee of $300 per person for 100 or fewer passengers (but more than 60 passengers). For each passenger over 100, the fare is decreased $2 per person for everyone (everyone byond the 100th).

What number of passengers would provide the maximum revenue?

------------------------------------------------------------------------------

The wording is tricky, and everything is relying on our interpretation.

It is easy to visualize the case where 60 < n <= 100: revenue R(n) = 300*n

For n > 100: R(n) = n*(300 - 2*(n - 100))

The above had to be VISUALIZED inside the mind.

"For each passenger over 100" is presented as (n - 100)

The two dollar rebate is, of course, -2

The 300 is obviously the $300 price tag for each ticket sold.

n is the number of tickets sold.

That was where the Schopenhauer-level thought-visualizing happens, but only after one has understood the exact meaning, the mathematical structure of this price-dependent revenue.

Once this is done, we might even robotically, that is, "mechanically" perform the operations, such as differentiation. We expand R(n) and find when R'(n) == 0.

That's not the Schopenhauer part, but the Newton/Leibniz part; although, Schopenhauerian perspectives come into play therein as well, since we have to understand/visualize the slopes to properly interpret the results of our derivations and computations.

I suspect that a more verbose expression of our own internal visualizations would be a warm welcome for the yet unborn students of mathematics and philosophy, and, I dare say, for literature itself.

___________________________________________________

R(n) = n*(300 - 2*(n - 100)) = n*(300 - 2n + 200) = n*(500 - 2n) = -2*n^2 + 500*n

Taking the derivative: R'(n) = -4*n + 500

Derivative R'(n) == 0 when -4*(n - 125) == 0 when n - 125 == 0 when n == 125

So the extremum is at n = 125

Since the second derivative R''(n) = -4 < 0, this extremum is a maximum.

Therefore the maximum revenue, if n > 100, is R(125) = 125*(500 - 2*125) = 125*(250) = 31250 dollars

If passengers is greater than 60 and less than or equal 100, then maximum revenue is at n = 100, so R(100) = 30000 dollars in that case. (trivial case)

___________________________________________________________

Do you see how visulaizing helped to derive the formula?

And yet, also, the understanding seems to be practically unnecessary in the process of differentiation. We mechanically work through the computation.

When it comes the results, we have to understand and apply the theorems, not so much by memorization, but by understanding the meaning of the second derivative, that is, the rate at which the slope is changing, the concavity. In this case, the concavity is downward so we can visulaize the peak as maximum.

If it were concave upward, the extremum would be at the bottom of a valley.

We are using mentally constructed visulaizations based upon our three-dimensional sensory experienced, although here we only need to visulaize two dimensions, which is easy for us.

If one did not know about second derivatives, one could use one's knowledge of the meaning of first derivatives, that is, slopes of curves.

If the extremum is at n = 125, test R'(124) and R'(126).

Since R'(n) = -4*(n - 125), R'(124) = -4*(-1) = 4 > 0

and R'(126) = -4*(1) = -4 < 0, so the slope increases to the left of n = 125, and the slope decreases to the right of n = 125, hence, we conclude, as with the second derivative test, that this is a maximum.

_______________________________________________

Note also that just in order to reflect upon such things as "word problems," I had to concentrate to block out my aging mother's continual litany of "what is on the Forgot Me List" [GROCERIES] ... So, we have to accept that life gets in the way of mathematics education. Sometimes you can gently express a need for your attention to be respected. That is, you have to demand the right to "use your own brain at the moment" (IF YA DON'T MIND!).

I have always saught, as a kind of Holy Grail, some unattainable Pure Intuitive Quasi-Instinctual Understanding of Mathematical Phenomena, and I might have fantasized that Schopenhauer possessed such an understanding. He may have found the symbolic coneptualized [FORMULATED] representation grotesque and technical, perhaps in a way analogous to ancient Hebrews finding any attempt to pronounce the Tetragrammaton YHWH offensive.

I'm not sure, but the Schopenhauer connection appears to be all about UNDERSTANDING. I have always romanticized what it might feel like to expereince a deepening of genuine understanding, rather than be mired down in the frustratingly tedius details of proof and code.

As I have aged, with my memory becoming even less sharp, I have had to fall back on VISUALIZATION in order to translate "words" into numerical representation.

This is mathematical thinking from the purely Schopenhauerian perspective, that is, in the privacy of the environment between our ears in honest contemplation and reflective consciousness, do we understand?

A textbook example where I apply Schopenhauerian Visualization would be appropriate. Holden suggested that Schopenhauer might hold the key to help the flies find our way out of the jar. I think that a key to this puzzle can be discovered in "word problems" --- where one must translate a verbally stated problem into some formula. That is the challenging part. Once that is formula, one might apply the necessary operations, such as discovering when the derivative is zero in Maximize/Minimize problems.

_______________________

Consider:

A chatering company will provide a plane for a fee of $300 per person for 100 or fewer passengers (but more than 60 passengers). For each passenger over 100, the fare is decreased $2 per person for everyone (everyone byond the 100th).

What number of passengers would provide the maximum revenue?

------------------------------------------------------------------------------

The wording is tricky, and everything is relying on our interpretation.

It is easy to visualize the case where 60 < n <= 100: revenue R(n) = 300*n

For n > 100: R(n) = n*(300 - 2*(n - 100))

The above had to be VISUALIZED inside the mind.

"For each passenger over 100" is presented as (n - 100)

The two dollar rebate is, of course, -2

The 300 is obviously the $300 price tag for each ticket sold.

n is the number of tickets sold.

That was where the Schopenhauer-level thought-visualizing happens, but only after one has understood the exact meaning, the mathematical structure of this price-dependent revenue.

Once this is done, we might even robotically, that is, "mechanically" perform the operations, such as differentiation. We expand R(n) and find when R'(n) == 0.

That's not the Schopenhauer part, but the Newton/Leibniz part; although, Schopenhauerian perspectives come into play therein as well, since we have to understand/visualize the slopes to properly interpret the results of our derivations and computations.

I suspect that a more verbose expression of our own internal visualizations would be a warm welcome for the yet unborn students of mathematics and philosophy, and, I dare say, for literature itself.

___________________________________________________

R(n) = n*(300 - 2*(n - 100)) = n*(300 - 2n + 200) = n*(500 - 2n) = -2*n^2 + 500*n

Taking the derivative: R'(n) = -4*n + 500

Derivative R'(n) == 0 when -4*(n - 125) == 0 when n - 125 == 0 when n == 125

So the extremum is at n = 125

Since the second derivative R''(n) = -4 < 0, this extremum is a maximum.

Therefore the maximum revenue, if n > 100, is R(125) = 125*(500 - 2*125) = 125*(250) = 31250 dollars

If passengers is greater than 60 and less than or equal 100, then maximum revenue is at n = 100, so R(100) = 30000 dollars in that case. (trivial case)

___________________________________________________________

Do you see how visulaizing helped to derive the formula?

And yet, also, the understanding seems to be practically unnecessary in the process of differentiation. We mechanically work through the computation.

When it comes the results, we have to understand and apply the theorems, not so much by memorization, but by understanding the meaning of the second derivative, that is, the rate at which the slope is changing, the concavity. In this case, the concavity is downward so we can visulaize the peak as maximum.

If it were concave upward, the extremum would be at the bottom of a valley.

We are using mentally constructed visulaizations based upon our three-dimensional sensory experienced, although here we only need to visulaize two dimensions, which is easy for us.

If one did not know about second derivatives, one could use one's knowledge of the meaning of first derivatives, that is, slopes of curves.

If the extremum is at n = 125, test R'(124) and R'(126).

Since R'(n) = -4*(n - 125), R'(124) = -4*(-1) = 4 > 0

and R'(126) = -4*(1) = -4 < 0, so the slope increases to the left of n = 125, and the slope decreases to the right of n = 125, hence, we conclude, as with the second derivative test, that this is a maximum.

_______________________________________________

Note also that just in order to reflect upon such things as "word problems," I had to concentrate to block out my aging mother's continual litany of "what is on the Forgot Me List" [GROCERIES] ... So, we have to accept that life gets in the way of mathematics education. Sometimes you can gently express a need for your attention to be respected. That is, you have to demand the right to "use your own brain at the moment" (IF YA DON'T MIND!).

For posterity, I will try to leave links to references.

From Lance Strate’s Lecture Notes on Teaching General Semantics (at www DOT generalsemantics DOT org):

Following Einstein’s non‐Newtonian physics, and non‐Euclidean geometry, Korzybski proposes a non‐Aristotelian mode of thought and communication.

Disclaimer: I place the above here as reference point. Having done that, I feel compelled to also note that, like Schopenhauer, Korzybski was kind of independent, and hence viewed as a cranky weirdo by the "insiders" of academia (The Church of Reason).

see: Why Korzybski Waned: Some Educated Guesses

Also, even though Korzybski represented "science," General Semantics itself has beeen viewed as "pseudo-philosophy" by W.V.O. Quine.

*The term is often used more casually to express contempt, irritation, or just dislike toward some idea or system of ideas. It is not, for the most part, used technically.*

Similarly, Arthur Schopenhauer wrote the following about Hegel:

"*If I were to say that the so-called philosophy of this fellow Hegel is a colossal piece of mystification which will yet provide posterity with an inexhaustible theme for laughter at our times, that it is a pseudophilosophy paralyzing all mental powers, stifling all real thinking, and, by the most outrageous misuse of language, putting in its place the hollowest, most senseless, thoughtless, and, as is confirmed by its success, most stupefying verbiage, I should be quite right.*" -- Arthur Schopenhauer, On the Basis of Morality, trans. E.F.J.Payne (Indianapolis: Bobbs-Merrill, 1965), pp.15-16.

*Schopenhauer's critiques of Hegel, Schelling, and Fichte are informed by his perception that their works use deliberately impressive but ultimately vacuous jargon and neologisms, and that they contained castles of abstraction that sounded impressive but ultimately contained no verifiable content. Soren Kierkegaard attacked Hegel in a similar manner, writing that it was pretentious for Hegel to title one of his books "Reality." To Kierkegaard, this indicated an attempt to quash critics even before criticism was voiced.*

Despite these attacks, Hegel is widely considered one of the most influential writers in world history: the rigor of his philosophy notwithstanding, Hegel had a significant impact on the writings of subsequent philosophers, for example Marx. Hegel scholar Walter Kaufmann contends that Schopenhauer's attacks actually illuminate more about Schopenhauer than about Hegel. Accusations that are similar in substance, if not in style, to Schopenhauer's have been made more recently against Martin Heidegger, postmodernists, and the adherents of French critical theory like Derrida, Jean Baudrillard, Julia Kristeva, Jacques Lacan and Lyotard.

Also studying Matrix.h code written by Stroustrup.

Thanks for your patience as I myself have been juggling contradictory ideas in this brain for years. The nature of the mind is chaotic. Systems will be left in ruins.

From Lance Strate’s Lecture Notes on Teaching General Semantics (at www DOT generalsemantics DOT org):

Following Einstein’s non‐Newtonian physics, and non‐Euclidean geometry, Korzybski proposes a non‐Aristotelian mode of thought and communication.

- A. Not anti‐Aristotle, but post‐Aristotle
- B. Aristotelian Laws of Thought (rules of logic)
- 1. Aristotle codified basic rules of logic, which we take for granted, seems natural to us
- 2. Law of identity. A=A. A thing is what it is. A man is a man, truth is truth. A is always A; A is all A
- 3. Law of excluded middle. A=B or A≠B, either/or. Either a man or not a man, either the truth or not the truth
- 4. Law of non‐contradiction. Not A=B AND A≠B. Not both a man and not a man, not both the truth and not the truth
- 5. Laws implies permanence, static relationship, polarization, things are discrete, not process. Allows us to categorize things (no double counting to confuse inventories).
- C. Korzybki’s Non‐Aristotelian Principles of Thought
- 1. Principle of Non‐Identity—A is not A. No identity relationships in nature. A map is not the territory it represents. The word is not the thing it represents. Whatever you say a thing is, it is not.
- 2. Principle of Non‐Allness—A is not all A. By labeling, we leave out information. A writer, fine, also a criminal.
- a. A map does not represent all of a territory. Words do not say all there is to say about the things they represent. A person cannot say all there is to say about a thing. The word “is” does no mean “equals.” Johnny is bad.
- b. Danger of absolutism, universalism
- 3. Principle of self‐reflexiveness. An ideal map would include a map of a map, etc. It is possible to speak words about words, and words about those words, etc. It is possible to react to one’s reactions, react to those reactions, etc. Statements about statements, evaluations of evaluations. Meta-communication, recursion.
- a. Ask—describe what you are doing right now. Like mirror reflecting mirror. Infinite. Mead, consciousness and self‐consciousness, imagine self as object, as others see us, and as we see them, etc. Carlyle, man is not unique because he uses tools, animals use tools. Man is unique because he uses tools to make tools. We use machines to answer and watch other machines. (VCR taping and not watching)
- b. Source of paradox. Barber shaves every man in the village who doesn’t shave himself, who shave the barber? To every rule there is an exception (as a rule itself). This statement is false.
- c. Whitehead and Russell, Theory of Logical Types, class cannot be a member of itself
- d. Gödel, Incompleteness Theorem
- e. Hofstadter, Gödel, Escher, Bach; self‐reflexiveness as basis of consciousness (self‐consciousness).

Disclaimer: I place the above here as reference point. Having done that, I feel compelled to also note that, like Schopenhauer, Korzybski was kind of independent, and hence viewed as a cranky weirdo by the "insiders" of academia (The Church of Reason).

see: Why Korzybski Waned: Some Educated Guesses

Also, even though Korzybski represented "science," General Semantics itself has beeen viewed as "pseudo-philosophy" by W.V.O. Quine.

Similarly, Arthur Schopenhauer wrote the following about Hegel:

"

Despite these attacks, Hegel is widely considered one of the most influential writers in world history: the rigor of his philosophy notwithstanding, Hegel had a significant impact on the writings of subsequent philosophers, for example Marx. Hegel scholar Walter Kaufmann contends that Schopenhauer's attacks actually illuminate more about Schopenhauer than about Hegel. Accusations that are similar in substance, if not in style, to Schopenhauer's have been made more recently against Martin Heidegger, postmodernists, and the adherents of French critical theory like Derrida, Jean Baudrillard, Julia Kristeva, Jacques Lacan and Lyotard.

Also studying Matrix.h code written by Stroustrup.

Thanks for your patience as I myself have been juggling contradictory ideas in this brain for years. The nature of the mind is chaotic. Systems will be left in ruins.

Schopenhauer on Laws of Thought:

Four laws

"The primary laws of thought, or the conditions of the thinkable, are four: – 1. The law of identity [A is A]. 2. The law of contradiction. 3. The law of exclusion; or excluded middle. 4. The law of sufficient reason." (Thomas Hughes, The Ideal Theory of Berkeley and the Real World, Part II, Section XV, Footnote, p. 38)

Arthur Schopenhauer discussed the laws of thought and tried to demonstrate that they are the basis of reason. He listed them in the following way in his On the Fourfold Root of the Principle of Sufficient Reason, §33:

A subject is equal to the sum of its predicates, or a = a.

No predicate can be simultaneously attributed and denied to a subject, or a ≠ ~a.

Of every two contradictorily opposite predicates one must belong to every subject.

Truth is the reference of a judgment to something outside it as its sufficient reason or ground.

Also:

The laws of thought can be most intelligibly expressed thus:

Everything that is, exists.

Nothing can simultaneously be and not be.

Each and every thing either is or is not.

Of everything that is, it can be found why it is.

There would then have to be added only the fact that once for all in logic the question is about what is thought and hence about concepts and not about real things.

— Schopenhauer, Manuscript Remains, Vol. 4, "Pandectae II", §163'

To show that they are the foundation of reason, he gave the following explanation:

Through a reflection, which I might call a self-examination of the faculty of reason, we know that these judgments are the expression of the conditions of all thought and therefore have these as their ground. Thus by making vain attempts to think in opposition to these laws, the faculty of reason recognizes them as the conditions of the possibility of all thought. We then find that it is just as impossible to think in opposition to them as it is to move our limbs in a direction contrary to their joints. If the subject could know itself, we should know those laws immediately, and not first through experiments on objects, that is, representations (mental images).

— Schopenhauer, On the Fourfold Root of the Principle of Sufficient Reason, §33'

Schopenhauer's four laws can be schematically presented in the following manner:

A is A.

A is not not-A.

X is either A or not-A.

If A then B (A implies B).

Two laws

Later, in 1844, Schopenhauer claimed that the four laws of thought could be reduced to two. In the ninth chapter of the second volume of The World as Will and Representation, he wrote:

It seems to me that the doctrine of the laws of thought could be simplified if we were to set up only two, the law of excluded middle and that of sufficient reason. The former thus: "Every predicate can be either confirmed or denied of every subject." Here it is already contained in the "either, or" that both cannot occur simultaneously, and consequently just what is expressed by the laws of identity and contradiction. Thus these would be added as corollaries of that principle which really says that every two concept-spheres must be thought either as united or as separated, but never as both at once; and therefore, even although words are joined together which express the latter, these words assert a process of thought which cannot be carried out. The consciousness of this infeasibility is the feeling of contradiction. The second law of thought, the principle of sufficient reason, would affirm that the above attributing or refuting must be determined by something different from the judgment itself, which may be a (pure or empirical) perception, or merely another judgment. This other and different thing is then called the ground or reason of the judgment. So far as a judgement satisfies the first law of thought, it is thinkable; so far as it satisfies the second, it is true, or at least in the case in which the ground of a judgement is only another judgement it is logically or formally true.

(From Wikipedia)

Four laws

"The primary laws of thought, or the conditions of the thinkable, are four: – 1. The law of identity [A is A]. 2. The law of contradiction. 3. The law of exclusion; or excluded middle. 4. The law of sufficient reason." (Thomas Hughes, The Ideal Theory of Berkeley and the Real World, Part II, Section XV, Footnote, p. 38)

Arthur Schopenhauer discussed the laws of thought and tried to demonstrate that they are the basis of reason. He listed them in the following way in his On the Fourfold Root of the Principle of Sufficient Reason, §33:

A subject is equal to the sum of its predicates, or a = a.

No predicate can be simultaneously attributed and denied to a subject, or a ≠ ~a.

Of every two contradictorily opposite predicates one must belong to every subject.

Truth is the reference of a judgment to something outside it as its sufficient reason or ground.

Also:

The laws of thought can be most intelligibly expressed thus:

Everything that is, exists.

Nothing can simultaneously be and not be.

Each and every thing either is or is not.

Of everything that is, it can be found why it is.

There would then have to be added only the fact that once for all in logic the question is about what is thought and hence about concepts and not about real things.

— Schopenhauer, Manuscript Remains, Vol. 4, "Pandectae II", §163'

To show that they are the foundation of reason, he gave the following explanation:

Through a reflection, which I might call a self-examination of the faculty of reason, we know that these judgments are the expression of the conditions of all thought and therefore have these as their ground. Thus by making vain attempts to think in opposition to these laws, the faculty of reason recognizes them as the conditions of the possibility of all thought. We then find that it is just as impossible to think in opposition to them as it is to move our limbs in a direction contrary to their joints. If the subject could know itself, we should know those laws immediately, and not first through experiments on objects, that is, representations (mental images).

— Schopenhauer, On the Fourfold Root of the Principle of Sufficient Reason, §33'

Schopenhauer's four laws can be schematically presented in the following manner:

A is A.

A is not not-A.

X is either A or not-A.

If A then B (A implies B).

Two laws

Later, in 1844, Schopenhauer claimed that the four laws of thought could be reduced to two. In the ninth chapter of the second volume of The World as Will and Representation, he wrote:

It seems to me that the doctrine of the laws of thought could be simplified if we were to set up only two, the law of excluded middle and that of sufficient reason. The former thus: "Every predicate can be either confirmed or denied of every subject." Here it is already contained in the "either, or" that both cannot occur simultaneously, and consequently just what is expressed by the laws of identity and contradiction. Thus these would be added as corollaries of that principle which really says that every two concept-spheres must be thought either as united or as separated, but never as both at once; and therefore, even although words are joined together which express the latter, these words assert a process of thought which cannot be carried out. The consciousness of this infeasibility is the feeling of contradiction. The second law of thought, the principle of sufficient reason, would affirm that the above attributing or refuting must be determined by something different from the judgment itself, which may be a (pure or empirical) perception, or merely another judgment. This other and different thing is then called the ground or reason of the judgment. So far as a judgement satisfies the first law of thought, it is thinkable; so far as it satisfies the second, it is true, or at least in the case in which the ground of a judgement is only another judgement it is logically or formally true.

(From Wikipedia)

I can see where the lines between philosophy and mathematics get blurry.

We do not need permission to study.

Even as many of us feel unworthy to explore certain disciplines, maybe due to what you mentioned about the territorialization of "mathematics-in-the-service-of-engineering-technology" by the Gorts-in-Charge.

There may be an artistic and highly personal engagement that takes place, where we each must be confused about our relationship with mathematics. Perhaps it is private. The gorts make one feel like they must be primed up by age 20 --- but what about a genuine decades long passionate engagement? Isn't that more in line with what Plato or Pythagoras would honor?

Ehh … Maybe modern society pays too much attention to the Greek philosophers and mathematicians, and not enough attention to the indigenous cultures' Old Wisdom.

Who knows? I can look at problems I was working on a just a few weeks ago, and become oppressed by the humbling realization that it still requires attention and THINKING. That is, problems have not become easier for me. I realize that I have to be in a very special mood just to "think mathematically" --- otherwise, yes, I am also mostly concerned with having shelter, food, clothing.

All my nephew has ever wanted was a small room and some space to work. Nearing 40, that is still something not so easy in Nordamerika [USA]. Like Kurt Vonnegut said, many people don't have a pot to piss in.

So, let us remember we are these apes who depend on food, apes who die of hypothermia or thirst. This is all I mean by ape, that one must recognize that our brains and our ability to inherit any kind of "culture" in the form of philosophy, mathematics, or science (or religion) depends first and foremost on the nourishment of our animal body.

So we ought not feel superior to those who have been prevented from inspecting books by the circumstances of their position in this insane asylum planet.

Peace.

**POST SCRIPTUM**

_______________________________________________________________

When I state that we are more ape than god, I mean that Schopenhauer, "the Lord Jesus Christ," and the Buddha are, of course, also more ape than god --- Mohammed as well? Of course. All animals who eat food and shit out foul smelling waste products.

When Schopenhauer advised us not to allow the mob to get a whiff of our humanity (that we are, after all, egotistical, vain and pitiable creatures, just like them), maybe he really meant for us to conceal our inner-ape, our demon.

My Inner Ape Killed Man. It's safer just to say, "I committed gorticide."

________________________________________________________________

The Steppenwolf killed Harry Hallar? What is the animal body hosting the social construction of Harry Hallar?

What is the animal body housing Man?

We do not need permission to study.

Even as many of us feel unworthy to explore certain disciplines, maybe due to what you mentioned about the territorialization of "mathematics-in-the-service-of-engineering-technology" by the Gorts-in-Charge.

There may be an artistic and highly personal engagement that takes place, where we each must be confused about our relationship with mathematics. Perhaps it is private. The gorts make one feel like they must be primed up by age 20 --- but what about a genuine decades long passionate engagement? Isn't that more in line with what Plato or Pythagoras would honor?

Ehh … Maybe modern society pays too much attention to the Greek philosophers and mathematicians, and not enough attention to the indigenous cultures' Old Wisdom.

Who knows? I can look at problems I was working on a just a few weeks ago, and become oppressed by the humbling realization that it still requires attention and THINKING. That is, problems have not become easier for me. I realize that I have to be in a very special mood just to "think mathematically" --- otherwise, yes, I am also mostly concerned with having shelter, food, clothing.

All my nephew has ever wanted was a small room and some space to work. Nearing 40, that is still something not so easy in Nordamerika [USA]. Like Kurt Vonnegut said, many people don't have a pot to piss in.

So, let us remember we are these apes who depend on food, apes who die of hypothermia or thirst. This is all I mean by ape, that one must recognize that our brains and our ability to inherit any kind of "culture" in the form of philosophy, mathematics, or science (or religion) depends first and foremost on the nourishment of our animal body.

So we ought not feel superior to those who have been prevented from inspecting books by the circumstances of their position in this insane asylum planet.

Peace.

___________________

When I state that we are more ape than god, I mean that Schopenhauer, "the Lord Jesus Christ," and the Buddha are, of course, also more ape than god --- Mohammed as well? Of course. All animals who eat food and shit out foul smelling waste products.

When Schopenhauer advised us not to allow the mob to get a whiff of our humanity (that we are, after all, egotistical, vain and pitiable creatures, just like them), maybe he really meant for us to conceal our inner-ape, our demon.

My Inner Ape Killed Man. It's safer just to say, "I committed gorticide."

________________________________________________________________

The Steppenwolf killed Harry Hallar? What is the animal body hosting the social construction of Harry Hallar?

What is the animal body housing Man?

I studied Categorical Logic,Preposition and Predicate logic, Truth tables ,logical operators,quantifiers-primarily to understand some of the modern philosophy better.

But it has helped me, as a side effect, to understand mathematics better too.

But it has helped me, as a side effect, to understand mathematics better too.

In our Burn Math Class thread, I cited an article which has great importance to me, for it allows me to embrace contradiction. It's only 8 pages, ah, but what a potent 8 pages it is! And the difficulty of tracking it down .... This is a breakthrough, although, I have no faith in systems of philosophy ... hardly even any faith in the foundations of mathematics, science, nor religion.

I wish to embrace the ruins. Please accept my truth. It is a truth which demands for accepting that a towering intellect may err in making comments on everything under the sun without accidentally exposing his ignorance.

Allow this thought to resonate:**Nothing could show better than the foregoing statement the scientific limitations of the otherwise towering intellect of Schopenhauer. Of the real difficulties that lie at the foundation of mathematics neither Goethe nor Schopenhauer had a true conception.**

From Goethe and Schopenhauer on Mathematics:

*Schopenhauer, like Goethe, did not appreciate at all what the French mathematical physicists had done. But how, without hardly any mathematical knowledge, could they expect to understand the Frenchmen? Nothing could show better than the foregoing statement the scientific limitations of the otherwise towering intellect of Schopenhauer. Of the real difficulties that lie at the foundation of mathematics neither Goethe nor Schopenhauer had a true conception. They were not able to anticipate even a possibility of the tremendous progress that has since been made and had been made during Schopenhauer's lifetime. But considered from a modern standpoint their often ill-tempered remarks appear as interesting flash-lights of a great historic period.*

The contradiction I embrace today is that genius and ignorance can go hand in hand, and that Schopenhauer's ignorance in some matters does not in any way disuade me from praising his otherwise towering character - the way he wrote against the Slave States, the hypocrisy he found in the Churches, the way he tore apart those institutions who expected to be off limits to criticism, his explorations into the metaphysics of sexual love, etc.

We must not demand perfection from anyone, not from Schopenhauer, and certainly not from ourselves.

NOTE: I would like to keep in mind what Holden wrote on page 3 of this thread:

Who are we? The Big Bad Real World would like us to feel unworthy of our own two cents. For good or ill, we have come to certain conclusions, a few dead ends ... yes, but at least we are trying to think as honestly as possible! I, for one, do not really expect to make sense of this life. We ask too much if we demand 100% coherency from ourselves or any other ape.

Let us remember that we are a species of ape. This might put things into better perspective for us.

I wish to embrace the ruins. Please accept my truth. It is a truth which demands for accepting that a towering intellect may err in making comments on everything under the sun without accidentally exposing his ignorance.

Allow this thought to resonate:

From Goethe and Schopenhauer on Mathematics:

The contradiction I embrace today is that genius and ignorance can go hand in hand, and that Schopenhauer's ignorance in some matters does not in any way disuade me from praising his otherwise towering character - the way he wrote against the Slave States, the hypocrisy he found in the Churches, the way he tore apart those institutions who expected to be off limits to criticism, his explorations into the metaphysics of sexual love, etc.

We must not demand perfection from anyone, not from Schopenhauer, and certainly not from ourselves.

NOTE: I would like to keep in mind what Holden wrote on page 3 of this thread:

Quote from: Holden

Thank you for answering my questions.First off, I want you to be clear that I hold your -maths- project in high esteem and it has rekindled my own desire to do more -math-.To attack it in anyway is the last thing I want to do.Having said that, I think you are talking of the following apparent paradox,that on one hand:

1.One considers Schopenhauer a great philosopher.

2.One accepts,for the most part,Schopenhauers world view.

3.One has come to believe that Schopenhauer somehow looks down upon -mathematics-.

On the other hand:

One spends a lot of time doing -mathematics-.

This is an apparent paradox,and channeling Wittgenstein,I suggest we do not try to resolve it,but that we dissolve it.

The dissolution will come about when one ponders the meanings of the words-Mathematics,Logic, and Philosophy and looks at them the way Schopenhauer used them. The way we use them.And most importantly the way the Gort uses them.

The apparent paradox will dissolve on its own when one considers the fact that our captains of the industry find it acceptable, nay,extremely useful,to teach -mathematics- in schools,(speaking only of the Indian Schools here, but I suspect that it might be true in your neck of the woods too), but NEVER logic( of any kind), NEVER philosophy(of any kind).

They teach- mathematics-, because they want clerks. In millions.They dont teach it because they want more Herr Hausers and Holdens. Maybe the way we are looking at -mathematics- ,was not the way Schopenhauer looked at it.Maybe he looked at it more radically .And maybe there is after all a non-gortish -mathematics-, which you practice and the gortish -mathematics-,which Schopenhauer was denigrating & is taught today in our schools.

Why do you think they lionise speed maths tricks? Why do they think mental maths is a big deal?

In the last analysis,it might be Schopenhauer himself who would show the fly the way out of the fly bottle.The gort says yes to Science, to Technology, to Engineering and to- Mathematics-.The gort laughs at ,or at best ignores logic and philosophy at the school level.There is something fishy here.

I am afraid the word -mathematics-,Deleuze would say, has been absolutely territorilized by the gort.

Maybe we need a new word,a better defined word. A deterritorilized word.

(No quotation marks in my keyboard so I am using -X- instead.I have used the word -math- in quotations everywhere as while I wish to use a better word,I am forced to use it due to the convention)

Who are we? The Big Bad Real World would like us to feel unworthy of our own two cents. For good or ill, we have come to certain conclusions, a few dead ends ... yes, but at least we are trying to think as honestly as possible! I, for one, do not really expect to make sense of this life. We ask too much if we demand 100% coherency from ourselves or any other ape.

Let us remember that we are a species of ape. This might put things into better perspective for us.

Somewhere in this thread I showed you how tables can be generated in Sage to help plot polar graphs. Now you can copy and paste it into http://sagecell.sagemath.org to see what I mean:

table([(i,n(cos(2*i), digits=4), cos(2*i)) for i in [0..2*pi+0.1, step=pi/12]], header_row=["$i$", "$cos(2*i)$", "$rational$"], frame=True)

___________________________________________________

t = var('t')

p1 = polar_plot(cos(2*t), t, 0, 2*pi)

p2 = polar_plot(sin(2*t), t, 0, 2*pi, linestyle="--")

show(p1 + p2)

_____________________________________________

table([(i,n(cos(2*i), digits=4), cos(2*i)) for i in [0..2*pi+0.1, step=pi/8]], header_row=["$i$", "$cos(2*i)$", "$rational$"], frame=True)

table([(i,n(cos(2*i), digits=4), cos(2*i)) for i in [0..2*pi+0.1, step=pi/12]], header_row=["$i$", "$cos(2*i)$", "$rational$"], frame=True)

___________________________________________________

t = var('t')

p1 = polar_plot(cos(2*t), t, 0, 2*pi)

p2 = polar_plot(sin(2*t), t, 0, 2*pi, linestyle="--")

show(p1 + p2)

_____________________________________________

table([(i,n(cos(2*i), digits=4), cos(2*i)) for i in [0..2*pi+0.1, step=pi/8]], header_row=["$i$", "$cos(2*i)$", "$rational$"], frame=True)

Quote

the requirement that the deceiver is capable of introducing deception even into mathematics ...

This reminds me of some comments made by Jason Wilkes in "Burn Math Class" during the process of coming up with "derivative formulas" where we "lie to Mathematics" in order to get the result we are looking for - and then reverse the lie at the end to make it true.

This is often called "factoring" in textbooks. We can pull anything out of anything. If we happen to want a c outside of (a + b), you end up writing "c * (a/c + b/c)".

This is a kind of deception.

I totally sympathize with those who say screw all this and get stoned drunk. I don't blame anyone for leaning in THAT direction.

Do you think we will reach a point where we won't even want to talk to anyone about anything?

Before I respond to your interesting post, I want to clarify what I suspect Schopenhauer means by the word, "difficult", at the end of the passage I cited.

I think Schopenhauer is referring to the god awful tediousness of the entire enterprise.

Although, when I am ardently engaged, the tedium is not so bad at all. It is only when I reflect upon how time consuming it is that I become discouraged. When I can transcend such concerns, I do not work in haste.

Now, on to your statements.

**1. [Mathematics] is not something which is completely distinct from what we call the world.**

It's a higher order of abstraction of it, whatever*it* is that we call "the world."

2. Mathematics dwells in the world as representation. Almost like this computer I am typing on.

Like the pen which is on my table ,right next of my hand.But the question is,what is a pen?

If I were to disappear this very moment would the pen continue to exist?No.

It is a part of what is called Maya( Maya is a word still used very often in India,it is by no means archaic).

I recognized Schopenhauer's "world as representation" as Maya as well. Even though it is not an idea presented through education, I had been exposed to the idea before reading Schopenhauer, and I was delighted that I had found a thinker who so forcefully supported the notion of ideality, that is, that there really is no such thing as the objective world, that we each exist in a life-world manufactured by our sensory apparatus.

I am glad you said the use of the term Maya is by no stretch of the imagination "archaic".

It is a supremely sophisticated concept.

3. I have to say I am in a pretty enviable position to appreciate where Schopenhauer is coming from - I mean being in India & all. What I heard as a kid tallies with what Schopenhauer has to say-about the world and about mathematics.

Didn't Schopenhauer himself anticipate exactly this somewhere in On the Will in Nature? I distinctly remember him predicting that his philosophy would be more readily embraced in the areas of the world whence came the doctrines that most influenced him. He was sickened by the simplistic views forced into the heads of the youth ... concerning religious indoctrination he saw as products of the realism inherent in the Judaic traditions in which Christianity and Islam are rooted.

Yes, you are fortunate to have been exposed to ideas that make Schopenhauer's views not at all extraordinary, but simply the truth.

I don't know if I ever mentioned that Alex Stepanov used to teach in India, and he was always encouraging his students there to ground themselves in their intellectually rich culture, and to try their best to resist being seduced by the vulgar materialism and wealth warped values of the "Bollywood" scene.

I do not claim to know India. I understand that it is vast and diverse. I even feel a little naive making general statements about it.

**4. Why my fixation with mathematics? I guess I want to understand it to as large as an extent as possible as for a very long time it has been a mystery to me.I see others around me struggling with it as well.The difference is that they focus raising their kid,I ,on understanding the world & mathematics.**

I have become more and more obsessed myself. I could fill many notebooks just going through the exercises in one textbook. I continue to go on tangents ... I call it a re-education campaign.

**5. Is life not frustrating? Then why do I assume that mathematics will not be frustrating. Is mathematics not a part of this world as well?**

The inherent frustrating aspects of exploring mathematics might have a lot to do with why many people write the subject off as something that they do not need to consider. For me, accepting this frustration, for example, knowing how much time can be spent just to become more familiar with notation and abbreviation symbols, is what enables me to devote my attention to it even without any long term goal in mind. In fact, it is better without any goals. Will-less contemplation.

6. no reply

**7.Mathematical equations are manifestations of the Will-to-Live.They are alive like me.They breathe too.They dwell in the realm of Maya.One has to accept the fact that by definition there is not sufficient reason for the existence of mathematical equations or the Will-to-Live.**

My existence is not necessary. The Fundamental Theorems of Calculus, Algebra, and Arithmetic are not necessary.

This world is unnecessary. There is no reason or ground explaining to us why the world is.

The existence of the world does not make any sense.

**8.The simplest of formulae and the most elaborate of proofs are equally mysterious.**

They are only as real as the dream I had last night.

Could this be the reason why, even after feeling one has mastered an area of mathematics, one can look at similar problems with the feeling that one has never studied it at all?

... Now I am sleepy ... for I am prone to fall asleep and lose all consciousness ... and all creatures sleep ... so this is some kind of shared universal illusion?

The knowledge does not seem quite solid. I mean, it can evaporate, so thinking we own our own knowledge is an illusion. We will deteriorate. Our intellect will lose memory. One might destroy all that knowledge with binge drinking as I have done for long periods of time throughout my life.

Presently I am going into a deep study of Polar Coordinates ... back to basics and beyond. I feel that the stretching of my mind will be a worthwhile project.

I want to go over analytic geometry, conic sections (in rectangular and polar coordinates).

I am ready to get down to the tedious details. I feel I owe it to myself.

The most important quality for me is honesty. For whatever reason, I find I struggle more with fundamentals than with supposedly more advanced topics. This is why I continually return to the basics to find the root of confusion. I am trying to really stay focused and to get a grip on this.

Quote from: Schopenhauer

If, therefore, we want to have abstract knowledge of space-relations, we must first translate them into time-relations, that is, numbers. For this reason, arithmetic alone, and not geometry, is the universal theory of quantity, and geometry must be translated into arithmetic if it is to be communicable, precisely definite, and applicable in practice. It is true that a spatial relation as such may also be thought in the abstract, for example "The sine increases with the angle," but if the quantity of this relation is to be stated, number is required. This necessity for space with its three dimensions to be translated into time with only one dimension, if we wish to have an abstract knowledge (i.e., a rational knowledge, and no mere intuition or perception) of space-relations--this necessity it is that makes mathematics sodifficult.

I think Schopenhauer is referring to the god awful tediousness of the entire enterprise.

Although, when I am ardently engaged, the tedium is not so bad at all. It is only when I reflect upon how time consuming it is that I become discouraged. When I can transcend such concerns, I do not work in haste.

Now, on to your statements.

It's a higher order of abstraction of it, whatever

2. Mathematics dwells in the world as representation. Almost like this computer I am typing on.

Like the pen which is on my table ,right next of my hand.But the question is,what is a pen?

If I were to disappear this very moment would the pen continue to exist?No.

It is a part of what is called Maya( Maya is a word still used very often in India,it is by no means archaic).

I recognized Schopenhauer's "world as representation" as Maya as well. Even though it is not an idea presented through education, I had been exposed to the idea before reading Schopenhauer, and I was delighted that I had found a thinker who so forcefully supported the notion of ideality, that is, that there really is no such thing as the objective world, that we each exist in a life-world manufactured by our sensory apparatus.

I am glad you said the use of the term Maya is by no stretch of the imagination "archaic".

It is a supremely sophisticated concept.

3. I have to say I am in a pretty enviable position to appreciate where Schopenhauer is coming from - I mean being in India & all. What I heard as a kid tallies with what Schopenhauer has to say-about the world and about mathematics.

Didn't Schopenhauer himself anticipate exactly this somewhere in On the Will in Nature? I distinctly remember him predicting that his philosophy would be more readily embraced in the areas of the world whence came the doctrines that most influenced him. He was sickened by the simplistic views forced into the heads of the youth ... concerning religious indoctrination he saw as products of the realism inherent in the Judaic traditions in which Christianity and Islam are rooted.

Yes, you are fortunate to have been exposed to ideas that make Schopenhauer's views not at all extraordinary, but simply the truth.

I don't know if I ever mentioned that Alex Stepanov used to teach in India, and he was always encouraging his students there to ground themselves in their intellectually rich culture, and to try their best to resist being seduced by the vulgar materialism and wealth warped values of the "Bollywood" scene.

I do not claim to know India. I understand that it is vast and diverse. I even feel a little naive making general statements about it.

I have become more and more obsessed myself. I could fill many notebooks just going through the exercises in one textbook. I continue to go on tangents ... I call it a re-education campaign.

The inherent frustrating aspects of exploring mathematics might have a lot to do with why many people write the subject off as something that they do not need to consider. For me, accepting this frustration, for example, knowing how much time can be spent just to become more familiar with notation and abbreviation symbols, is what enables me to devote my attention to it even without any long term goal in mind. In fact, it is better without any goals. Will-less contemplation.

6. no reply

My existence is not necessary. The Fundamental Theorems of Calculus, Algebra, and Arithmetic are not necessary.

This world is unnecessary. There is no reason or ground explaining to us why the world is.

The existence of the world does not make any sense.

They are only as real as the dream I had last night.

Could this be the reason why, even after feeling one has mastered an area of mathematics, one can look at similar problems with the feeling that one has never studied it at all?

... Now I am sleepy ... for I am prone to fall asleep and lose all consciousness ... and all creatures sleep ... so this is some kind of shared universal illusion?

The knowledge does not seem quite solid. I mean, it can evaporate, so thinking we own our own knowledge is an illusion. We will deteriorate. Our intellect will lose memory. One might destroy all that knowledge with binge drinking as I have done for long periods of time throughout my life.

Presently I am going into a deep study of Polar Coordinates ... back to basics and beyond. I feel that the stretching of my mind will be a worthwhile project.

I want to go over analytic geometry, conic sections (in rectangular and polar coordinates).

I am ready to get down to the tedious details. I feel I owe it to myself.

The most important quality for me is honesty. For whatever reason, I find I struggle more with fundamentals than with supposedly more advanced topics. This is why I continually return to the basics to find the root of confusion. I am trying to really stay focused and to get a grip on this.