The Tortoise and Achilles don't agree...

[Plane]

... that logic requires logic, or that logic springs from nature ,or that logic is logical.


Or perhaps I don't understand?

https://en.wikipedia.org/wiki/What_the_Tortoise_Said_to_Achilles

[Xavier_Onassis]

This is called Zeno's paradox, and it is a logic problem built on a faulty logic, in which Achilles can never pass the tortoise, he can only get closer and closer to it. He cannot even get even with it.

In reality, Achilles can easily pass the tortoise. I think that some form of calculus can explain mathematically how the tortoise can be passed, but the math that explains how Achilles can pass the tortoise mathematically had not been invented.

[Plane]

  This relates to Zeno's paradox , but while Calculus solved the problem that Zeno describes in measurement and time, the problem of accepting standard axioms still exists.

  Suppose you and I argue as these character's do.

  One of us maintains that if two things are equal to each other , then a third thing equal to one of them is equal to both.

   The other of us argues that this is a special case that may not be true in all of reality.

    The first of us points to a real situation in which this is so, that if two sides of a triangle are the same length and their angle at apex is the same as another triangle then these two triangles must cover the same area.

      The second could argue that this is only true in equal topography, that a triangle laid on the salt flat will include a much much less surface area than the same measure taken on a mountainous area.

[Xavier_Onassis]

The second could argue that this is only true in equal topography, that a triangle laid on the salt flat will include a much much less surface area than the same measure taken on a mountainous area.


The second person would be correct, as a third dimension was added, that of depth.

[Plane]

Euclid does not like the second persons argument.

On equal topography the rule holds true , but does the rule become useless without the special case of equal topography?

Euclid seems to make the assumption that measurements of two dimensional shapes are made on two dimensional surfaces, additional rules are required for third dimensions.

Failing to posit this , or failing to accept it as stated, spoils the Argument.

[Xavier_Onassis]

I took way more geometry than I needed, but I never read Euclid.

[Plane]

If you took any geometry at all , then you have been exposed to Euclid.

Advances beyond Euclid are mostly pretty recent, he was the last word for a long stretch.

[Xavier_Onassis]

I am sure that the authors of my HS geometry books were familiar with Euclid. But I have not read Euclid himself.

Zeno's paradox was not covered in either my Geometry I or Geometry II class. That would have added too much excitement, perhaps.

My Geometry teacher and Algebra teacher was a women named Irene LaFrenz. She was a typical German (Alsatian) woman in that she had only one way to do everything.
Each theorum we roved had to be proven in the exact steps she gave us, and then we had to draw a box around the answer. The best way to really impress her was to draw the box with a straightedge. Drawing an oblong or anything curvy was frowned upon. Half the time I could not do the problems in the proper number of steps: instead of five, I would do four, or six. This was repeated in Algebra II.

[Plane]

http://aleph0.clarku.edu/~djoyce/java/elements/elements.html


   That is too bad, geometry can be fun.

    And, ... there are usually many ways to complete a solution , else none.
     The single correct way is, in terms of math, a myth.

   It is not at all too late to read the original.

     Knowing now that some of the Axioms have been modified for special purpose , even so, all of the "elements" still work and the proofs are still elegant and useful.
    Twenty four centuries ago and still useful stuff.

[Xavier_Onassis]

Geometry can certainly be useful. even though I have had little need for it beyond rug measurement and linoleum calculations.

I don't think the idea of learning stuff being fun ever occurred to any of my HS teachers. I had a shop teacher who made me proud of my talent at  making a able lamp.

[Plane]

I had a shop teacher who made me proud of my talent at  making a able lamp.

I had much the same experience in shop.

I bet that this shop experience taught you a little geometry while you weren't looking.

Euclidian geometry applies directly to sheet metal work, carpentry and a lot of other crafts.

Some people don't enjoy this as much in abstract as they do in hardware.

Me for example.