On Jul 11, 2013, at 1:10 AM, Joe Niederberger <email@example.com> wrote:
> By your argument, these things were not really be known before they could be proved by some modern standard.
No. I am saying that they are not really known before they are known. You are being sloppy with the word "known".
Let's take your example of the MVT. If I draw a continuous curve on the board from point A to point B it is obvious that if I trace this curve with my finger or just with my eyes that it must go through every point on it's path. But that is just the reality of the world we live in. What thought or reasoning is involved with this? Where is the logical argument or conclusion at the end of this? You are trying to equate our concrete sense of the world with mathematical reasoning. I say they are two very different things.
You are also confusing what I am saying with rigor. I don't care about rigor. I also don't care if it is geometric or algebraic. All I care about is backed with REASON.
Appealing to one's rat-sense of continuous is not the same thing as appealing to one's abstract sense and theory of continuity. That is my line in the sand. That is my warning to those who become enthralled with seeing mathematics in a picture and forget that they knew the mathematics FIRST.
You seem to be insisting that one can have a mathematical visual experience without reasoning. Yet every time I ask you to explain it, out pours the reasoning.
Here is a thought experiment for you. Take a person with no exposure whatsoever to mathematics and shown them the continuous curve. And then you take a look at the continuous curve. I guarantee that both of your visual rat-sense experiences will be the same. So, I ask you, are you both having a mathematical experience or not? Do you have a line in the sand?