Author Topic: Schopenhauer's Philosophy of Mathematics  (Read 6888 times)

0 Members and 0 Guests are viewing this topic.

Holden

  • { ∅, { ∅ } }
  • Posts: 3792
  • Hentrichian Philosophical Pessimist
Schopenhauer's Philosophy of Mathematics
« on: July 10, 2016, 12:26:54 pm »
First off,I relinquish any claim to have come up with a new philosophy of mathematics, my earlier posts may have given such an impression –what I believe in is Schopenhauer’s philosophy of mathematics.I know that in the recent past there has been one book by a philosophaster wherein it is claimed that Schopenhauer never recognized a math genius .
The said philosophaster also claimed that Schopenhauerian “Idealism” is faulty and argued for realism. I can see plainly that the philosophaster never read all the works of Schopenhauer.

I greatly doubt if even Nietzsche read everything by Schopenhauer.As for me,I have taken a solemn oath of reading every single word Schopenhauer ever wrote ,if it’s the last thing I do.
Now, day before yesterday,I was reading Parerga and Paralipomena (Vol.I).
I was reading “Fragments for the History of Philosophy”.In that essay when Schopenhauer talks about Pythagoras,he as says that the basis of his theory of music is mathematics. I think the philosophaster never read that.

There it is.
I summarise Schopenhauer’s mathematical philosophy thus:
1.Mathematics is synthetic apriori
2.Mathematics is ruled by the principle of sufficient reason of being
3.Mathematics must be studied in the spirit of “denial of the will-to-life”
4.Mathematics is as calming as music

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.

https://youtu.be/o6O_d1DVk3U
« Last Edit: July 10, 2016, 12:42:25 pm by Holden »
La Tristesse Durera Toujours                                  (The Sadness Lasts Forever ...)
-van Gogh.

Share on Facebook Share on Twitter


Kaspar Hauser

  • { }
  • { ∅, { ∅ } }
  • Posts: 4299
  • Life teaches me not to want it.
    • What Now?
Re: Schopenhauer's Philosophy of Mathematics
« Reply #1 on: July 15, 2016, 10:56:08 am »
I am going through WWRv1, using the index of WWRv2 (Payne translations).   There is one bookmark in WWRv1, and it is just an index card with page numbers which reference mathematics.  The numbers are faded.  First of all, I will rewrite the numbers on a fresh index card, and then list those pages here similar to the way I listed References to Qualitas Occulta.

Over not too long a period of time, we ought to have most such references collected here, and then it might be a good idea to use httrack to store the site on a hard drive or, better still, a flash drive.   I'll peck away at it.

These page numbers correspond to the Payne translation (Dover edition), and will not match up with the full online text referenced above.  To reference the pages listed, use this link to the Dover edition/Payne translation for WWRv1

50, 54, 81, 85, 95-6, 121, 144, 189, 222, 247, 342, 346, 431, 449, 465, 469, 480

Euclid (52, 55, 63, 438)

and WWRv2.

34, 72, 85, 89, 106, 121, 143, 195, 328, 379

Euclid (3, 130)

Late last night I was reflecting on some of what he said, and it is uncanny how it related directly to the issues my own brain is dealing with as far as the differences between numeric, algebraic, and geometric/graphical representations are concerned.


Quote
I remember having read in the introduction to a German translation of Euclid that we ought to make all beginners in geometry draw the figures first before proceeding to demonstrate, since they would then feel geometrical truth, before the demonstration brought them complete knowledge. 

I am placing some excepts of what I was focusing on in the middle of the night.

Quote
Here another peculiarity of our faculty of knowledge comes under discussion, and one that could not be observed previously, until the difference between knowledge of perception and abstract knowledge was made perfectly clear.  It is that the relations of space cannot directly and as such be translated into abstract knowledge, but only temporal quantities, that is to say numbers, are capable of this. Numbers alone can be expressed in abstract concepts exactly corresponding to them; spatial quantities cannot.

Quote
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 so difficult.

[H: This is what makes mathematics so difficult, the translation of geometric understanding into algebraic and numerical understanding].

This becomes very clear when we compare the perception of curves with their analytical calculation, or even merely the tables of the logarithms of trigonometrical functions with the perception of the changing relations of the parts of a triangle expressed by them. What vast tissues of figures, what laborious calculations, would be required to express in the abstract what perception here apprehends perfectly and with extreme accuracy at a glance, namely how the cosine diminishes while the sine increases, how the cosine of one angle is the sine of another, the inverse relation of the increase and decrease of the two angles, and so on! How time, we might say, with its one dimension must torture itself, in order to reproduce the three dimensions of space! But this was necessary if we wished to possess space-relations expressed in abstract concepts for the purpose of application. They could not go into abstract concepts directly, but only through the medium of the purely temporal quantity, number, which alone is directly connected to abstract knowledge.  Yet it is remarkable that, as space is so well adapted to perception, and, by means of its three dimensions, even complicated relations can be taken in at a glance, whereas it defies abstract knowledge, time on the other hand passes easily into abstract concepts, but offers very little to perception. Our perception of numbers in their characteristic element, namely in mere time, without the addition of space, scarcely extends as far as ten.

Note the exclamation points.

Quote
Beyond this we have only abstract concepts, and no longer perceptive knowledge of numbers. On the other hand, we connect with every numeral and with all algebraical signs precise and definite abstract concepts. 

Incidentally, it may here be remarked that many minds find complete satisfaction only in what is known through perception. What they look for is reason or ground and consequent of being in space presented in perception. A Euclidean proof, or an arithmetical solution of spatial problems, makes no appeal to them. Other minds, on the contrary, want the abstract concepts of use solely for application and communication. They have patience and memory for abstract principles, formulas, demonstrations by long chains of reasoning, and calculations whose symbols represent the most complicated abstractions. The latter seek preciseness, the former intuitiveness. The difference is characteristic.

Much of what Schopenhauer is pointing out comes into play when working with vectors.  They have direction and magnitude, and hence are represented algebraically with numbers ... quantities, and yet they are spatial.

For instance, when visualizing the projection of one vector on another, there are formulas which you plug the numerical components into to yield numerical values of the resultant vector ... and then, if you are so inclinied, you can use the numerical values of corresponding components to graph the vectors, component-wise, which is when you "see" it.  This is the kind of thing Schopenhauer is talking about when screaming exclamations.

What is interesting is that one can find the angle between two vectors even if the dimension of the vectors is higher than 3, in other words, even when our faculty of knowledge is incapable of imagining n-dimensional space.  We can accomplish it in the abstract with algebra and arithmetic, but we cannot spatialize, or geometricize the algebra at that point.

One thing that becomes clear early on when first exploring differential equations is that many equations cannot be solved by quantitative methods, and we have to use qualitative methods.   

Direction Fields: Qualitative Analysis
Separation of Variables:  Quantitative Analysis
Approximation Methods: Numerical Analysis


Schopenhauer's discussions about the nature of the particular characteristic difficulties involved in mathematical "knowing", especially how he points out the interconnected relationship between algebraic and geometric representation is comforting because he is discussing mathematics from a higher ground.  He held philosophy to be deeper than all the sciences because he did not take things for granted.  With mathematics we always take time and space as given. 


There is something very comforting about having Schopenhauer discuss these limitations, for then we begin to recognize where some of our "difficulties in understanding" come from, the nature of these difficulties.   In other words, these are not personal, individual defects, but a priori limitations.  Once we become aware of this, we might come to experience perplexity differently.
« Last Edit: July 16, 2016, 09:28:04 am by Nothing »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

Holden

  • { ∅, { ∅ } }
  • Posts: 3792
  • Hentrichian Philosophical Pessimist
I am sorry I could not write sooner.I have been studying Schopenhauer's thoughts on mathematics.
I finished reading this essay called" Transcendent Speculation on the Apparent Deliberateness in the Fate of the Individual."
It helped me greatly in understanding how I came across you.
You asked me if there are any old students in India.I think there are very few such people here.Every man here gets married in the late twenties or early thirties.And then kids invariably follow.All relatives look at me as a freak.They think I care about what they think about me.
They are greatly mistaken.

Your other question was whether we should focus on whats happening in the world or not-wars,killings etc.
I think we should ignore them completely.We can do nothing about it whatsoever. The roots of evil are far deeper than these politicians are willing to accept.I could just read Schopenhauer for the rest of my life while the world around me continues to burn.
At present I am reading the Essay on Spirit Seeing.I think Schopenhauer was a genius greater than a zillion Einsteins put together.
As far as programming is concerned I will certainly take it up.When I am reading Schopenhauer I feel as if I have left the mortal world all together. One of the biggest lies which Schopenhauer exposes is that the world(or math for that matter) is logical.
When in reality it is just all our intuitive perception.

 
La Tristesse Durera Toujours                                  (The Sadness Lasts Forever ...)
-van Gogh.

Kaspar Hauser

  • { }
  • { ∅, { ∅ } }
  • Posts: 4299
  • Life teaches me not to want it.
    • What Now?
Re: Schopenhauer's Philosophy of Mathematics
« Reply #3 on: July 17, 2016, 10:08:26 pm »
Quote from: Holden
All relatives look at me as a freak.They think I care about what they think about me. 

They are greatly mistaken.

I'm sure my relatives see me as some kind of freak as well, but, then again, wasn't Schopenhauer a freak?   I am the oldest male cousin.  My maternal grandparents had 8 children, 6 of which were daughters, all now grandmothers themselves.   So, I am apparently one of the few of the many cousins who has not reproduced, and those couple others who haven't are still young and most likely striving to do so.  It no longer concerns me.  The children of my cousins will soon be throwing more puppies into this, sure enough.  It is not my place to discourage them.  They wouldn't listen to me anyway.  I must be a bit of a joke, eh?

To each his own.  I have no ill will.

I try not to judge anyone, but I certainly envy no one.

Quote from: Holden
Your other question was whether we should focus on whats happening in the world or not-wars,killings etc.

I think we should ignore them completely.We can do nothing about it whatsoever. The roots of evil are far deeper than these politicians are willing to accept.I could just read Schopenhauer for the rest of my life while the world around me continues to burn.

True.  We can do nothing about it.  No people on earth have a monopoly on evil.  As you have noted many times, it is a matter of chance that anyone has the temperament they have.  I do not believe holding signs in the air will change the order of things.  I heard "Be the change you want to see in the world."   

I want to nurture intelligence.  I do not want to see intelligence at the mercy of brutality. 

Quote from: Holden
At present I am reading the Essay on Spirit Seeing.I think Schopenhauer was a genius greater than a zillion Einsteins put together.

Me too.  I try to remember this when I get discouraged about not being an Einstein when it comes to physics.  You know, I love the algebraic calculations and formulas, all that.  I'm just not a very fast thinker.  Sometimes, though, I think this might be the ideal pace for such things.  One can't read formulas as literature.  Still, I respect Schopenhauer's grade of intelligence.  It may sound like an oxymoron, but it seems as though Schopenhauer had a high degree of intelligence in his heart, what modern psychologists might even call "emotional intelligence."

I would go so far as to say that it would require emotional intelligence to discern that splitting "the atom" might not be such a bright idea.  That's why I don't worship science or "scientists" even as I myself am very fascinated with science, especially Queen Mathematics.  I jokingly referred to my grandfathers as "servile scientists".  They did as they were told, and were proud of their golden handcuffs.  I can still hear my paternal grandfather teasing me (when I was a confused and rebellious teenager), "What are you going to do, live in a tent?"

He would say things like, "The world doesn't owe you a living."

The maternal grandfather, also a scientist (an engineer) would encourage me to think with my head and not with my heart.  Often, throughout my life, I have thought with my heart.

Someone who is thinking with their heart is not likely to help create weapons of mass destruction for the State calling it his "duty".   

"It's either that, or they will send me to the front lines!"

"Umm, one could be a conscientious objector."

Excuse me, I went off on tangent.   

I was just considering that just as there are different grades of will,  there may be different kinds of intelligence.  Who is to say why certain employees are promoted to certain positions of authority.  It may have little to do with intelligence or aptitude and more to do with attitude and servility.

Soldiers do not get praised for thinking.

One thing is certain, marching soldiers are not thinking. 

Hence, in order to promote patriotism or nationalism or corporatism, a certain kind of thinking must be viewed as subversive, criminal, or at least "diseased".
« Last Edit: July 20, 2016, 12:23:20 am by {∅} »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

Kaspar Hauser

  • { }
  • { ∅, { ∅ } }
  • Posts: 4299
  • Life teaches me not to want it.
    • What Now?
Re: Schopenhauer's Philosophy of Mathematics
« Reply #4 on: July 27, 2016, 10:39:29 am »
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 so difficult.

Holden, remember when you asked me if I look at a curve in a book before drawing it?  I will explain the general scenario.  What I quoted from Schopenhauer [WWRv1] is very relevant.

Suppose I am asked to find the area of the inner loop of the polar curve r = 1 + 2*sin(theta), where theta is the angle in polar coordinate system.

What I generally do first is use a computer algebra system such as Sage or even gnuplot (or a graphing calculator) to graph the polar curve from theta = 0 to theta = 2*pi.

What I have found helpful is to display a table evaluating r as a function of theta, and then using the table to plot the points on a graph I draw by hand, a polar grid where the circles represent distance from the pole.  I plot the points and connect the dots.

Were I to attempt to just sketch the curve on the fly, this is when the curve looks skewed.  This has everything to do with what Schopenhauer wrote nearly 200 years ago.  As he was fond of saying, we are living in the self-same world.  Even with the assistance of technology, we are still dealing with the fundamental nature of the relationship between time and space, which is manifest in the relation between algebra/arithmetic and geometry, the relationship between numbers and geometric objects.

"geometry must be translated into arithmetic if it is to be communicable" ~ S

In other words, I can't just mechanically apply the formula for finding the area of a polar curve, but have to first analyze the graph to decide upon the appropriate interval, the limits of integration.

It's kind of fascinating to witness the exact nature of what Schopenhauer is talking about.  You know, for many years I did not pay much attention to his statements about arithmetic and geometry, and their direct relation to his statements about time and space.  I have been mostly influenced by his thoughts on the denial of the will, the wickedness of optimism, etc.

Schopenhauer has been called "the artist's philosopher," but what he wrote about mathematics is quite profound.  I am currently quite enthusiastic about this bridge between algebra and geometry that used to be called "analytic geometry".

Where I used to race into the mechanical operations involving calculus, I am finally beginning to take my time and really analyze the situation slowly and calmly, creating graphs, tables, taking advantage of computer algebra systems graphing capabilities, but also jotting down the table of values next to a hand sketched graph.  Maybe I have finally matured enough to take my time with each exercise I engage with.

PS:  Even though seeing the computer generated graph and finding the angle theta where r is zero, I could also skip all the visualization and find this by algebraic means, what Schopenhauer calls "with arithmetic".

In fact, I prefer analyzing it this way with pencil and paper before proceeding with the visualization process (geometric representation).

You just ask yourself, "When is r = 0?"

r = 0 when 1 + 2*sin(theta) = 0

... when 2*sin(theta) = -1

... when sin(theta) = -1/2

theta = arcsin(-1/2)

This occurs when theta = pi + pi/3 = 7*pi/6

Also when theta = -pi/6

Note that 2*pi + 7*pi/6 = 4*pi/6 + 7*pi/6 = 11*pi/6, which is the same angle as -pi/6

At this point, as is the scenario frequently, one could find oneself drawing a graph of sin on a rectangular coordinate system to note where sin(x) = -1/2

The depth and the degree of analysis depends on just how much one really wants to grasp the described situation that the curve represents.

Remember that all this analysis takes place even before applying any calculus to determine the area of the "inner loop".  When working in haste, this stage might be done abruptly, whereas as once the definite integral is set up, one considers the calculus involved as the "real part" of the problem solving activity.   I confess that this is how my brain attacked such problems for years.

For whatever reasons I am a little calmer now and take some pleasure in the analysis stage that appears to be boundless, limited only by something nudging you to "make progress", "forge ahead."

God forbid one should go off on a tangent!

How does one balance this tension?  I mean, on the one hand, you want to understand things deeply, and on the other hand, there is a kind of pressure to not spend "too much time" on the task at hand.

So, when looking at r = 1 + 2*sin(theta), it helps to look at this with both polar AND rectangular coordinates, where r(theta) = 1+2*sin(theta) translates to f(x) = 1 + 2*sin(x).  This shows visually what I initially found algebraically.

It seems to be a matter of "extracting intelligence" by whatever means necessary, and there are no laws dictating precisely how to go about extracting such intelligence.  Maybe I was the type of student who wanted a fool-proof step by step procedural algorithm so I could compute and calculate in a mechanical manner.  Now that I am no longer in an environment (i.e., "school") where memorization and stressful fast calculations were demanded (exams), and since I no longer despair about "transforming from a janitor into a 'scientist' ", I can at long last enjoy the process of gaining intelligence through exploration, inspection, and in depth analysis.

At this juncture I come to realize that I do not have to choose between polar or rectangular coordinates, but am free to use both simultaneously since each gives its own valuable information in the "intelligence gathering" phase of the "problem solving" process.

It only took me over 20 years to realize this, although, 20 years ago, I actually was not even stretching my mind enough to consider polar coordinates and rectangular coordinates being used in harmony.  I told you I was a slow learner. It's as though not even a great teacher from a distant {exotic) land could have taught me this.  I had to learn it in the spirit of exploration, in a state of mind attained through what the industrious refer to as "goofing off."

Speaking of exotic lands, with all the commotion over politics and religion all over the world, it is kind of revealing that like-minded individuals from diverse cultures can discuss mathematical truths as autonomous subjects, thereby transcending ego and ethnicity.

Might a shared fascination with the art of mathematics demolish the socially constructed masks dr-ape-d over our subjectivities? 

Contemplation of sine and cosine occur in an inner realm that transcends the boundaries of states and nations ... and is truly "apolitical".

Returning to the very beginnings of The WWRv1 in search of anything the great genius Schopenhauer had to say about mathematics has had an effect on my approach.  No longer do I feel like an old Steppenwolf struggling with school-boy mathematics exercises, but embrace each set of problems and each textbook I engage with in the spirit of the ancient scholars who who studied with in an exploratory manner, and not simply to have gold stars to put on their job resume.

I am not a man who prays to any gods, but I find myself saying as like Sioux medicine man Black Elk prayed, "Make me to walk in a sacred manner."

I want to explore mathematics in a sacred manner.  Schopenhauer pointed out exactly why mathematics can be difficult, and so we might embrace these difficulties as inherent in the limits we face in translating geometrical representations into algebraic representations.  May we proceed with humility and patience rather than arrogance and frustration.
« Last Edit: July 28, 2016, 10:49:43 pm by {∅} »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

Kaspar Hauser

  • { }
  • { ∅, { ∅ } }
  • Posts: 4299
  • Life teaches me not to want it.
    • What Now?
Re: Schopenhauer's Philosophy of Mathematics
« Reply #5 on: July 28, 2016, 09:43:26 pm »
FOOTNOTE

simple example of table generated by sage command:
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)



+----------+------------+--------------+
| $i$      | $cos(2*i)$ | $rational$   |
+==========+============+==============+
| 0        | 1.000      | 1            |
+----------+------------+--------------+
| 1/12*pi  | 0.8660     | 1/2*sqrt(3)  |
+----------+------------+--------------+
| 1/6*pi   | 0.5000     | 1/2          |
+----------+------------+--------------+
| 1/4*pi   | 0.0000     | 0            |
+----------+------------+--------------+
| 1/3*pi   | -0.5000    | -1/2         |
+----------+------------+--------------+
| 5/12*pi  | -0.8660    | -1/2*sqrt(3) |
+----------+------------+--------------+
| 1/2*pi   | -1.000     | -1           |
+----------+------------+--------------+
| 7/12*pi  | -0.8660    | -1/2*sqrt(3) |
+----------+------------+--------------+
| 2/3*pi   | -0.5000    | -1/2         |
+----------+------------+--------------+
| 3/4*pi   | 0.0000     | 0            |
+----------+------------+--------------+
| 5/6*pi   | 0.5000     | 1/2          |
+----------+------------+--------------+
| 11/12*pi | 0.8660     | 1/2*sqrt(3)  |
+----------+------------+--------------+
| pi       | 1.000      | 1            |
+----------+------------+--------------+
| 13/12*pi | 0.8660     | 1/2*sqrt(3)  |
+----------+------------+--------------+
| 7/6*pi   | 0.5000     | 1/2          |
+----------+------------+--------------+
| 5/4*pi   | 0.0000     | 0            |
+----------+------------+--------------+
| 4/3*pi   | -0.5000    | -1/2         |
+----------+------------+--------------+
| 17/12*pi | -0.8660    | -1/2*sqrt(3) |
+----------+------------+--------------+
| 3/2*pi   | -1.000     | -1           |
+----------+------------+--------------+
| 19/12*pi | -0.8660    | -1/2*sqrt(3) |
+----------+------------+--------------+
| 5/3*pi   | -0.5000    | -1/2         |
+----------+------------+--------------+
| 7/4*pi   | 0.0000     | 0            |
+----------+------------+--------------+
| 11/6*pi  | 0.5000     | 1/2          |
+----------+------------+--------------+
| 23/12*pi | 0.8660     | 1/2*sqrt(3)  |
+----------+------------+--------------+
| 2*pi     | 1.000      | 1            |
+----------+------------+--------------+

attached is the simple graph generated by the sage command:

 t = var('t')
polar_plot(cos(2*t), t, 0, 2*pi)

If I am looking for the area inside two curves, it helps to plot both, and then find the intervals of integration.

Too time consuming?  Computers help, but when I am calm, I like to draw the polar coordinates and plot them, coloring the areas of interest.

sage: p1 = polar_plot(cos(2*t), t, 0, 2*pi)
sage: p2 = polar_plot(sin(2*t), t, 0, 2*pi, linestyle="--")
sage: show(p1 + p2)

At the point where I begin the integration, this is what Schopenhauer is referring to when he says, "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 so difficult."

While computing the integral there are lots of opportunities for arithmetic errors.  I always check my results with a computer algebra system.

The last thing I want to think about when engrossed in such activity is "what will I be studying in the autumn or by the time winter returns or before I am 50."

While my shelves are filled with a lifetime's worth of material to explore, I can only focus on the task at hand.  It does take some discipline to stick with the textbooks and not skip through tedious exercises.

Could you imagine a janitor locking himself in his supervisor's office to "play with graphs" on the computer?

I am the rebel slave who has been rewarded with freedom from my atrocious behavior.   :)

FREEDOM!

PS:  The following tables were more helpful.  All I did was change the step size to pi/8.  I did this after algebraically solving cos(2*t) = sin(2*t).  t = pi/8

That's where the two curves intersect.   Do you see how the algebraic and numeric are so important in communicating the geometric?  I think, no, I know this is exactly the kind of thing Schopenhauer was referring to.

When you look at the graphs and then look at the numbers in the tables representing the positions of the curves, think of Schopenhauer's statements.

sage:  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)
+---------+------------+--------------+
| $i$     | $cos(2*i)$ | $rational$   |
+=========+============+==============+
| 0       | 1.000      | 1            |
+---------+------------+--------------+
| 1/8*pi  | 0.7071     | 1/2*sqrt(2)  |
+---------+------------+--------------+
| 1/4*pi  | 0.0000     | 0            |
+---------+------------+--------------+
| 3/8*pi  | -0.7071    | -1/2*sqrt(2) |
+---------+------------+--------------+
| 1/2*pi  | -1.000     | -1           |
+---------+------------+--------------+
| 5/8*pi  | -0.7071    | -1/2*sqrt(2) |
+---------+------------+--------------+
| 3/4*pi  | 0.0000     | 0            |
+---------+------------+--------------+
| 7/8*pi  | 0.7071     | 1/2*sqrt(2)  |
+---------+------------+--------------+
| pi      | 1.000      | 1            |
+---------+------------+--------------+
| 9/8*pi  | 0.7071     | 1/2*sqrt(2)  |
+---------+------------+--------------+
| 5/4*pi  | 0.0000     | 0            |
+---------+------------+--------------+
| 11/8*pi | -0.7071    | -1/2*sqrt(2) |
+---------+------------+--------------+
| 3/2*pi  | -1.000     | -1           |
+---------+------------+--------------+
| 13/8*pi | -0.7071    | -1/2*sqrt(2) |
+---------+------------+--------------+
| 7/4*pi  | 0.0000     | 0            |
+---------+------------+--------------+
| 15/8*pi | 0.7071     | 1/2*sqrt(2)  |
+---------+------------+--------------+
| 2*pi    | 1.000      | 1            |
+---------+------------+--------------+


sage:  table([(i,n(sin(2*i), digits=4), sin(2*i)) for i in [0..2*pi+0.1, step=pi/8]], header_row=["$i$", "$sin(2*i)$", "$rational$"], frame=True)
+---------+------------+--------------+
| $i$     | $sin(2*i)$ | $rational$   |
+=========+============+==============+
| 0       | 0.0000     | 0            |
+---------+------------+--------------+
| 1/8*pi  | 0.7071     | 1/2*sqrt(2)  |
+---------+------------+--------------+
| 1/4*pi  | 1.000      | 1            |
+---------+------------+--------------+
| 3/8*pi  | 0.7071     | 1/2*sqrt(2)  |
+---------+------------+--------------+
| 1/2*pi  | 0.0000     | 0            |
+---------+------------+--------------+
| 5/8*pi  | -0.7071    | -1/2*sqrt(2) |
+---------+------------+--------------+
| 3/4*pi  | -1.000     | -1           |
+---------+------------+--------------+
| 7/8*pi  | -0.7071    | -1/2*sqrt(2) |
+---------+------------+--------------+
| pi      | 0.0000     | 0            |
+---------+------------+--------------+
| 9/8*pi  | 0.7071     | 1/2*sqrt(2)  |
+---------+------------+--------------+
| 5/4*pi  | 1.000      | 1            |
+---------+------------+--------------+
| 11/8*pi | 0.7071     | 1/2*sqrt(2)  |
+---------+------------+--------------+
| 3/2*pi  | 0.0000     | 0            |
+---------+------------+--------------+
| 13/8*pi | -0.7071    | -1/2*sqrt(2) |
+---------+------------+--------------+
| 7/4*pi  | -1.000     | -1           |
+---------+------------+--------------+
| 15/8*pi | -0.7071    | -1/2*sqrt(2) |
+---------+------------+--------------+
| 2*pi    | 0.0000     | 0            |
+---------+------------+--------------+
sage:


[attachment deleted by admin]
« Last Edit: October 25, 2020, 11:22:39 am by Sticks and Stones »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

Holden

  • { ∅, { ∅ } }
  • Posts: 3792
  • Hentrichian Philosophical Pessimist
The Deus Deceptor
« Reply #6 on: July 29, 2016, 11:12:21 am »
That’s very  thorough reading of Schopenhauer’s theory of mathematics, I mean the distinction between the geometrical and the arithmetical.  For the last few days I could not write to you & I am sorry for that,but apart from working in the salt mines I also was lost in deep thought.
Turns out I have something rather original to say about mathematics.I mean,I have not yet read everything by Schopenhauer( I am on my way),so maybe I will come across a similar thought in his works somewhere but not to put too fine a point on it,this is the theory I have constructed:
1.   Mathematics is a part of the world. Its not apart from it.One could say that it’s the reflection of the world even.At any rate, it is not something which is completely distinct from what 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).

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.
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.
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?
6.   An Illustration: One morning as I get ready to go office I realise that my bike key is not where it should be.I panic.I search everywhere but I cannot find it.I am late for the office.I am in pain.
What causes that pain? Do I think that the “key” is without an iota of will? I have come to realise that not just so called living beings but everything is a manifestation of the will-to-live.Everything.
Even this keyboard I am typing on,even the ceiling fan where is rotating over my head.The tubelight.Everything.
So,when I lose the key I have two options:
a.I keep looking of the key.I may find it ,I may not.If I select this option I am being Nietzschean.
Even if I find the key today,will I be able to find it tomorrow as well?
b.I let the key go,after looking at the few likely places.I get a duplicate key made.

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.
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.
9.The reason why I fail to comprehend math is this: There is not sufficient reason for its existence & I look of it.
There is absence of sufficient reason for the existence of the Will-to-Live and mathematical equations.
10.I look at my keyboard,I wonder what is it made of.Plastic.And how to they manufacture plastic? One could go on & on this way. Why is a^2-b^2= (a+b)(a-b)?No final proof here.Or anywhere.I can never understand this equation the way I want to,because there is absence of sufficient reason for its existence. It just exists. But I know what’s hidden behind the equation.
The blind,hungry & irrational Will.Existing without Sufficient Reason.

Suppose tomorrow I wake up & find that when I put two red ball together with two blue balls I am getting five ball and not four.Will be surprised?
As surprised as I am to get four balls.
There is no reason why it should be four and not five.It might as well be 217.

 Schopenhauer wrote: "This world could not have been the work of an all-loving being, but that of a devil, who had brought creatures into existence in order to delight in the sight of their sufferings."

The deus deceptor (French dieu trompeur) "deceptive god" is a concept of Cartesianism. In the First Meditation where Descartes stated that he supposed not an optimal God but rather an evil demon "summe potens & callidus" ( "most highly powerful and cunning").
The progression through the First Meditation, leading to the introduction of the concept of the evil genius at the end, is to introduce various categories into the set of dubitables, such as mathematics (i.e. Descartes' addition of 2 and 3 and counting the sides of a square). Although the hypothetical evil genius is never stated to be one and the same as the hypothetical "deus deceptor," (God the deceiver) the inference by the reader that they are is a natural one, and the requirement that the deceiver is capable of introducing deception even into mathematics is seen by commentators as a necessary part of Descartes' argument. Scholars contend that in fact Descartes was not introducing a new hypothetical, merely couching the idea of a deceptive God in terms that would not be offensive.
« Last Edit: July 29, 2016, 11:13:59 am by Holden »
La Tristesse Durera Toujours                                  (The Sadness Lasts Forever ...)
-van Gogh.

Kaspar Hauser

  • { }
  • { ∅, { ∅ } }
  • Posts: 4299
  • Life teaches me not to want it.
    • What Now?
Re: Schopenhauer's Philosophy of Mathematics
« Reply #7 on: July 29, 2016, 04:37:43 pm »
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.

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 so difficult.

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.
« Last Edit: July 29, 2016, 11:48:10 pm by {∅} »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

Kaspar Hauser

  • { }
  • { ∅, { ∅ } }
  • Posts: 4299
  • Life teaches me not to want it.
    • What Now?
Re: Schopenhauer's Philosophy of Mathematics
« Reply #8 on: August 04, 2016, 10:28:04 am »
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? 
« Last Edit: August 04, 2016, 10:56:04 am by {∅} »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

Kaspar Hauser

  • { }
  • { ∅, { ∅ } }
  • Posts: 4299
  • Life teaches me not to want it.
    • What Now?
Re: Schopenhauer's Philosophy of Mathematics
« Reply #9 on: June 18, 2017, 05:38:26 pm »
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)

« Last Edit: June 18, 2017, 05:40:28 pm by Raskolnikov »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

Kaspar Hauser

  • { }
  • { ∅, { ∅ } }
  • Posts: 4299
  • Life teaches me not to want it.
    • What Now?
Re: Schopenhauer's Philosophy of Mathematics
« Reply #10 on: April 28, 2020, 11:22:07 am »
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:

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.
« Last Edit: April 28, 2020, 11:53:41 am by Demon Spirit »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

Holden

  • { ∅, { ∅ } }
  • Posts: 3792
  • Hentrichian Philosophical Pessimist
Re: Schopenhauer's Philosophy of Mathematics
« Reply #11 on: April 28, 2020, 02:47:11 pm »
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.
La Tristesse Durera Toujours                                  (The Sadness Lasts Forever ...)
-van Gogh.

Kaspar Hauser

  • { }
  • { ∅, { ∅ } }
  • Posts: 4299
  • Life teaches me not to want it.
    • What Now?
Re: Schopenhauer's Philosophy of Mathematics
« Reply #12 on: April 28, 2020, 07:34:02 pm »
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? 
« Last Edit: April 29, 2020, 02:57:00 pm by Demon Spirit »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

Holden

  • { ∅, { ∅ } }
  • Posts: 3792
  • Hentrichian Philosophical Pessimist
Re: Schopenhauer's Philosophy of Mathematics
« Reply #13 on: April 29, 2020, 05:58:59 am »
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)
La Tristesse Durera Toujours                                  (The Sadness Lasts Forever ...)
-van Gogh.

Kaspar Hauser

  • { }
  • { ∅, { ∅ } }
  • Posts: 4299
  • Life teaches me not to want it.
    • What Now?
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.
  • 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.

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.
« Last Edit: April 29, 2020, 04:17:44 pm by Demon Spirit »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~