Thursday, April 2, 2009
Invented or discovered?
Shall we start the famous 'is mathematics discovered or invented' debate anew? I remember hMensa posed it earlier. The question is, does nature display some kind of mathematical properties or is mathematics a product of our imagination (or ingenuity) that happen to describe nature? In class many of my students were not impressed when I tried to explain that there is good justification to say that mathematics is a discovery like the sciences. Does nature speak the language of mathematics or it is just us?
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This question actually did come up in my class also and as I believed then, mathematics as we know it now is combination of discovery and invention.
Ultimately, the very essence of mathematics pervades nature from the four leave clover to the golden ratio, the divine proportion which is a representation of logic and order, in the same way math tries to make sense of things through the logical method. Therefore if we said math was invented, then one can say science is invented - and it's not(of that I'm sure)
On the other hand,the very value of pi exists and can be discovered but the value we attribute to it is invented. Certain things are invented to aid us better understand (Am I right in my reasoning?)
The question of whether we were getting closer o the truth also came up- the search for the value of the infinite prime number etc.- and I think (in reference to an earlier post by S.Kidane) that if we did discover the exact value of pi, the infinite prime number, the ultimate truth, the universe would either self destruct or evolve into something more incomprehensible.
Math as a 'language'? A global communicative tool,yes but as far as TOK is involved I don't think so.
What do you think? Are getting closer to the truth,can we get there and if we do get there what would happen?
When we were discussing this in class Mr. Kidane said mathematics was a discovery because math is displayed in nature. I strongly disagree.
I think math is purely an invention. Let's take the the golden ratio. Before you even think of dividing the length of a rectangle by its width you have to have defined the concepts of division, length and numbers. All these are purely invented from the mind. This is mathematics. The fact that nature displays some mathematical symmetry or pattern doesn't make it discovered.
Again, in rabbit reproduction Fibonacci sequence that we discussed, we had to have some idea as to what a sequence is. That is the math - the definition and invention of sequences.
We also talked about Kepler's laws of planetary motion particularly his 3rd law of periods. The law is stated as follows, "The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit." The proportionality mentioned here is again defined previously in mathematics before it can be applied to the situation.
I think another thing that makes people take a view contrary to mine is what we perceive math to be. If i were to write an equation down on paper, that is not math. The math is in your head, it is the thought. The equation that is written is just a tool to enable others "see" what you are thinking.
So, unless of course you tell me nature gives you the thought, then mathematics in my view, is not discovered.
I hope someone would convince me otherwise. I particularly like this question. Let's keep the discussion going.
The statement of Kepler's 3rd law was taken from:
http://hyperphysics.phy-astr.gsu.edu/HBASE/Kepler.html#c6
Mathematics is a tool or model used to describe parts of reality. Did mathematics exist before the first mathematicians? I think not. What existed was the thought as Gartaa previously said. It then must have been invented. What was discovered was how to make these models fit reality.
Something is said to be discovered if it is already around us but we are not familiar to it or something which is already created but not known. And invention is more like creating new thing that do not actually exist. Math is some how it is invention and it is more of discovery. The basic tools in math like the ideas of axioms and postulates are invented and other things are discovered based on the inventions. This means invention has lead to discovery.
I like what is going on here.Some of the questions raised are really serious. In line with Chilot's post I think it might be helpful to clarify our understanding of the word 'discovery'. Once we have a common understanding it would be easier to go to the next step. If I invent an equation and the equation describes some truth about the sun, isn't that equation the property of the sun whether I 'invent' the equation or not?
I think I have an interesting response for Gartaa.
Issac, Kimi, Okwase and I had a similar argument about math's discovery or invention a night ago (till like 12 midnight...) annyyywaayyysss...
Basically, the problem with defining whether or not math is invented comes from symbolism.
Math is a humanly comprehensible symbol expressing how nature works. From things like Fibernacci's sequence etc. etc. it is clear that the universe indeed follows a certain order, or way of doing things.
All we did was to call this order math, dubbing what nature did to turn poop into manure, a 'function', since one thing goes in and another goes out. Sequences, since we noticed that if you count the number of petals on flowers of different age, you notice that they drop by 1. How are you going to define nature's action without a symbol for it?? Such is math!!
Since nature performs actions, yielding resultant numerical quantities, we therefore have no choice but to describe those actions and DEFINE them. But by defining them, the absolute value does not change. By dubbing seeing how many times one quantity can enter another division, we are not loosing the essence of the action, we just need a way of saying 'finding out how many times one value cam..' you see how long I sound? So we called it division.
We CANNOT communicate without symbolism, because there is no way of expressing a certain event if we do not name it. e.g F=ma in any other way. You have to assign symbols to the fact that Force is equal to mass * acceleration. If you have a way of describing this without applying symbols, I will applaud you. :) iIn explaining dont use English, cos if u do you are using mass to represent how much pull is on the object by the earth. And by saying earth you are using a symbol for the spherical thing we are standing on, and when you say sphere you are using a symbol for the shape of what we are standing on, and when you say standing you mean a symbol for... do you see what i mean?
I understand what Gartaa is saying, but it is IMPOSSIBLE to efficiently and productively refer to something without giving it a name. Just because we put names to the concepts of nature (i.e divisioin, multiplication, etc.) does not change the fact that the ABSOLUTE VALUE, or the essence of what we are saying is exactly the same.
This is why I say math is discovered, but the symbols for it are created. Because in another universe the symbol for the numerical value of 1 is (@*), but you cannot tell me that if I but 1 beer, and the alien buys one dgeidwciwopdqzo( alien beer) we are not talking about the same numerical value. We are! But because, like me the alien has to NAME what he is doing, he then called 1 (@*). So does this mean he invented math? NO! He just uses different symbols to express the same quantity...
Also, in relation to what LJwella said, we could not possibly make our models fit reality, because our models had experiment and observation as a basis, so they were DERIVED from real situations. At least in some cases... if you look at how shells are taught in IG chemistry, the math there is absolute bullocks, coz we know that there arent even properly defined shells.
Also in relation to what Mr. Kidane said, if you think you invented an equation, but it applies for every situation you defined it for, then that is no invention. It is a discovery but you do not know. :)
Hope this helps, open to other views. Love this question but have some VERY strong views about it!
Vladimir,
I think that you do not fully understand what I said (NOT BECAUSE YOU DON'T AGREE WITH ME). Before getting into that, I would like to say that giving names to things is not defining them. If I give something the name "csdhcn" it does not tell me what it is. Again, as I said before, math is not the symbols it is the thought. If you think of dividing it is the concept of dividing that is the math.
Now, to what I said, nature may be seen to follow some mathematical behaviour. Think about this. Read the next sentence slowly. It...follows...mathematical....behaviour. Which means the mathematical behaviour was there and it follows it.
Hence, mathematics is not discovered but invented
I'd just like to state the following alternatives for the record. It may be that some of you would like to reflect further, and this might be a skeleton on which to build the tissues.
It is hard to refute that the physical universe in which we live SEEMS to exhibit features that are related to mathematics. For a convincing example (out of many), take Kepler's 3rd law.
What does this mean? There are at least three alternatives:
1) It tells us something about the world (physical universe) - namely that the world is in a deeply fundamental sense mathematical in nature
2) It tells us something about mathematics - namely that mathematics is a profoundly effective tool for reaching truth and understanding about the universe
3) It tells us something about ourselves as humans - that we have a predilection for things mathematical
If claim 1 is correct, perhaps we could derive important truths about the universe simply from doing mathematics.
If claim 2 is correct, then perhaps empirical data can be used to derive new truths about the world on a mathematical basis (ie we need to start with "the world" but mathematics allows us to discover more about it without necessarily doing experiments at every stage)
If claim 3 is correct, maybe this is a particularly troubling example of "confirmation bias" - namely that we look for quantifiable and numerically-tractable aspects of the world and over-emphasize their importance.
Now what do you think?
In an attempt to coalesce the alternatives, I think Mathematics is a reflection of the NATURE of the human mind. To keep this post as concise as possible, I'd say that by nature, humans (reference to claim 3) "have a predilection for things mathematical" and therefore looks out for trends, patterns, sequences or anything mathematically oriented in nature. And the world we perceive has been so kind as to display these trends, patterns, sequences or anything mathematically oriented.
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