Pathagorean Theorem

Take an arbitrary square. Along the top side measure a distance a across from the upper-left corner. Using the same length measure down from the upper-right corner, across to the left from the lower-right corner, upward from the lower-left corner. Label all four of the measured distances a (equal by construction). Label the rest of each side b (equals subtracted from equals are equal). Connect the points between a and b of adjacent sides creating four congruent trangles (two sides and included angle). Since they are congruent the third sides of each triangle are equal. Label these sides c. The interior area can be shown to be a square. We have already show the sides, c, to be equal. The angles are straight angles (180 degrees) less the sum of the two acute angles of right triangles (90 degrees). Now we use the fact that the area of the whole is equal to the sum of the areas of the parts. The area of the whole is side squared = (a + b)^2. The area of the center square is c^2. The area of each corner triangle is one-half the base times the height which is axb/2, so four of them have area 2xaxb. Therefore

(a + b)^2 = c^2 + 2xaxb.

(a + b)^2 - 2xaxb = c^2

a^2 + 2xaxb + b^2 - 2xaxb = c^2

a^2 + b^2 = c^2

Or the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.

Ok, I skipped a couple of proofs. Construct a rectangle with base width w and height h. Draw a diagonal creating two congruent triangles (two sides and included angle). Thus two times the area of the triangle is the area of the rectangle making the area of one triangle wxh/2.

Now consider a corner of the rectangle from which the diagonal is drawn. It is a right angle divided into two angles each of which is an acute angle of a right triangle. This shows that the sum of the acute angles of a right triangle are a right triangle. It doesn't matter that these acute angles are actually from different triangles because the two are congruent.

Conciousness

Conciousness is not understood nor, in my opinion, understandable. What physical process could possibly create awareness? Could a robot (or a computer) be concious? If it were how would we know? Conciousness is not the same as intelligence. Machines can be intelligent. People speculate on whether dogs or other animals are concious. I believe they are but have no way of knowing. The only conciousness I can be sure of is my own. It is unique to me and to my lifetime. We have an enormous amount of memory stored in our brains. We may not be able to describe in detail our drive to work but we notice if something changes. The amount of memory we are aware of at one time is very limited. I have no awareness of anything that happened before I was born and don't expect any after I die. Suppose I hadn't been born. After all, the chances of my having been must be infintessimal. And even more improbable is my being alive today. My lifetime is not even a blink in the age of the earth. Suppose time could be reset. I wish I were 19 again. If I were I wouldn't know anything about the future. Time is the only one of Einstein's four dimensions that is unidirectional. That is supposedly guaranteed by the law of increasing entropy.

MIZAR

MIZAR is a formal language that enables proof-checking.  The most important applications of MIZAR today are playing the role of a proof assistant to support creating rigorous mathematics, in mathematics education, and in software and hardware verification.

So how much mathematics can you teach a computer?