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!).