I recall a student asking me what the “square root of two” (hereafter called √ 2) meant. I was about to say that it was a number that when squared you got 2 but that seemed a little too glib so I held my tongue and told her that I would think about it and tell her tomorrow.
The more I thought about it the more I realized that her question was not trivial. √2 is an infinite, non-repeating decimal but what does this mean. To make a long story short, I decided that √2 did not exist in the Real World.
In my state of ‘eureka’ I told a colleague that √2 didn’t exist in the Real World. He said that of course it did. It was the length of the diagonal of the unit square. I then asked him, When was the last time you saw a unit square.” Unit squares aren’t in the Real World either.
When teaching about the graphs of functions I would take a function, say f(x)= x+3, and plot a few points, talk about slopes and intercepts and points and eventually draw a line on the blackboard. Then I would say, “This is the graph of the line f(x) = x+3.”
But it wasn’t the graph of f(x) = x+3. It was a picture of the graph of f(x) = x+3. The graph of a function is a set of points and points aren’t in the Real World. I can’t say that a dot on a blackboard is a point. It is a picture of a point.
I liken the picture of a graph to the picture of a unicorn. The picture of a unicorn is not a unicorn. It is the picture of something not in the Real World.
And here is one of my main points. The symbol is not the object. The symbol is a representation, a picture if you will, of the object.
The proof that √2 is not a rational number is not a proof that √2 exists, just that if √2 did exist it wouldn’t be a rational number.
Here I think the Greeks went astray. Previous to the proof that √2 wasn’t rational the Greeks had assumed all numbers were rational. Since √2 wasn’t rational they concluded that there were other kinds of numbers that weren’t rational instead of concluding that √2 didn’t exist.
In my calculus book I introduce the “Ideal World” and put √2 in the Ideal World along with infinity. Unit squares are in the Ideal World. Graphs are in the Ideal World. Unicorns are in the Ideal World. Real World problems are modeled in the Ideal World and solved in the Ideal World. The answer is interpreted in the Real World and then experiments are done to see if the answer makes Real World sense. The pure mathematician stays pretty much in the Ideal World and if the Ideal World laws are obeyed she thinks the problem is solved. It is left for the applied mathematicians and scientists to see if her answer makes sense.
So what is in the Real World and what is in the Ideal World? I seem to be edging perilously close to Platonism except that I don’t think the Ideal World exists. I think the Ideal World is a construction of humanity. I wonder at my colleagues who are atheists but believe in √2.
What about functions? I think functions as we know them are human constructs made in the Ideal World. Our concept of function is that of a single valued function. This arose because to a first approximation our world is single valued. A rock always falls 16 feet in the first second. Note that I am already in the Ideal World. When we teach mathematics, we don’t say that the rock falls about 16 feet in about 1 second. In the Real World nothing is exactly 16 feet in length and no interval of time is exactly 1 second.
None the less, when I tune my radio to 89.9 I get KUNM every time. We believe in the repeatability of physical laws and hence single valued functions.
But in the world of the dolphin there is no repeatability. If you drop a rock repeatedly in a swimming pool it doesn't fall to the bottom in a single time; not even close. So the dolphin wouldn't develop single valued functions. Functions are a human construct.
I read where an applied mathematician said that the only numbers in the Real World are finite decimals. But a decimal point followed by a billion zeros followed by a 1 is a finite decimal. To a mathematician it is a perfectly good positive number. In the Real World it equals 0. Mathematicians let Δx go to zero without a qualm, even in calculus courses for business majors, even if x represents light bulbs. What does a billionth of a light bulb look like?
Personally I put very, very, very small numbers in the Ideal World. Likewise with very big numbers, I put 10 raised to the billionth power in the Ideal World. I am using extreme values and a number slips into the Ideal World a long time before 10 raised to the billionth power.
I think the student wished she hadn't asked about √2.
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I understand you argument but I am not to sure if I agree. The theorem of Pythagoras is often used to prove that the mysterious number √ 2 existed in the real world. People often say that it surly existed we get triangles that are 1 meter by 1 meter and are right angled triangles in the real world and surly the hypotenuse is equal to √ 2. However, you use the clever but sneaky argument that the triangle its self cannot existed in the real world. There seems to be little someone can do to argue against you. Distance its self cannot be measured. Even if you believe in the plank distance (the smallest-ed possible distance possible), which I don't by the way, you run into a major problem. 1 plank distance by one plank distance triangle cannot have a √2 hypotenuse due to the fact that a plank distance can only be in whole numbers, since it is the smallest possible unit after all. This is one of the reasons why I find the plank length theory so silly. Even if you represent a million plank lengths as a distance the answer will not be equal to √2. If you do not believe in the plank length then you cannot even say that it is 1m by 1m. And all of this is only true if you have a perfected triangle. In the real word a perfected triangle does not existed. The side will never be perfectly straight and the angle is never perfectly 90 degrees. So lets see what can be said about your world. The hypotenuse in the real world is a irrational number. In fact everything in the real world that cannot be a a integers has to be a irrational number. This because they never perfectly fit a defined unit. Another key point is that you brought up infinity and compared it to √2. There is a big difference ({infinity - √2} ignore the joke). √2 could in theory existed in the real world because a triangle can be 1m by 1m in theory but in practice it cannot. Infinity on the other hand cannot existed in theory or in practice. Therefore, you cannot use the argument that √2 is a theoretical in the same way you would use that infinity is theoretical. By this you must conclude that the number can existed given in the real world given the right circumstances. Although you may think it will never happen but look at the size of the universe and then consider the time in the universe. (yes, i do relies the question was does it existed not has it or will it existed but your point of view seems to be that it never has and never will). The theorem of Pythagoras is not the only theorem that lead to √2 being the answer. You have to consider the large amount of theorems that could lead to this answer that have not been discovered yet. With the theorems we know there are simply to many factors but there is a good chance that there may one day arise a idea that will need only the 2 lengths to be equal. Something like Pythagoras but just without the perfect angle and line. A loop hole. With the way that nature works it is possible for two things to be the same length when given enough time and space. If enough of these form there is a good chance that they may arrange themselves in such a way that a certain section of them will actually equal √ 2. Therefore according to the possibilities it most likely has existed or is going to existed. Even for a small amount of time. Even if you disagree with what I am saying think about time. There has to be √2 seconds. Surly. There has to be something in the universe that shrinks or grows. if it grows from 1m to 2m does it not mean that it must have reaches √2 m in some point in time in order to get to 2m.
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