Author Topic: Schopenhauer's Philosophy of Mathematics  (Read 6881 times)

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Kaspar Hauser

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Re: Schopenhauer's Philosophy of Mathematics
« Reply #15 on: May 05, 2020, 08:36:58 am »
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.
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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?
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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.
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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)
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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.
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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!).   >:(

 ;)

« Last Edit: May 05, 2020, 10:12:54 am by Heinrich von Hauser »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

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Holden

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Re: Schopenhauer's Philosophy of Mathematics
« Reply #16 on: May 06, 2020, 02:22:28 am »
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",""))
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Holden

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Re: Schopenhauer's Philosophy of Mathematics
« Reply #17 on: May 06, 2020, 04:29:51 am »
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.
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Holden

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Holden

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On Badiou ( For Herr Hauser)
« Reply #19 on: May 08, 2020, 05:11:33 pm »
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”
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Holden

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Re: Schopenhauer's Philosophy of Mathematics
« Reply #20 on: May 11, 2020, 01:42:44 pm »
"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)
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Kaspar Hauser

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Re: Schopenhauer's Philosophy of Mathematics
« Reply #21 on: October 24, 2020, 07:27:26 am »
I am reading through this paper on neologicism left by Holden.

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.
« Last Edit: October 24, 2020, 08:55:07 pm 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 ~

Kaspar Hauser

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Re: Schopenhauer's Philosophy of Mathematics
« Reply #22 on: October 25, 2020, 12:21:55 pm »
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?
« Last Edit: October 25, 2020, 07:48:59 pm 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

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Kaspar Hauser

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Re: Schopenhauer's Philosophy of Mathematics
« Reply #24 on: December 04, 2020, 10:48:32 am »
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?"
« Last Edit: December 04, 2020, 11:13:34 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

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Re: Schopenhauer's Philosophy of Mathematics
« Reply #25 on: December 05, 2020, 09:28:07 am »
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.
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Holden

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Re: Schopenhauer's Philosophy of Mathematics
« Reply #26 on: May 21, 2021, 03:50:51 am »
"Where there is matter, there is geometry." ~ Johannes Kepler
La Tristesse Durera Toujours                                  (The Sadness Lasts Forever ...)
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