I felt a great burden lift from my shoulders. We have differing numbering systems as well. However, if we are discovering something deeper and more profound about the universe itself it could imply that mathematics is unbound, and has no end to revelations it can provide. Therefore this question can be rephrased. I mean, the best mathematics is an art. I do think mathematics is also discovered. Or the interminable time it took mankind to introduce a zero into arithmetic? Those that marvel at the ubiquity of mathematical applications have perhaps been seduced by an overstatement of their successes.
In that case, then Kant is not saying that we project math onto anything. But there may be intelligences that do not count. If you have a negative number and you square it, you get a positive number. The absolutely pure invention is not as interesting from that perspective. This intimately linked geometry and algebra to be two sides of the same coin and allowed mathematicians the algebraically analyse the world around them which, naturally, is full of geometrical shapes leading to immense breakthroughs which would otherwise not have been possible. Imagine something as fundamental as Pi falling under copyright.
In all instances of discovery and invention there are both choices to make and unalterable truths to uncover. Russell employed a version of set theory in his reconstruction of mathematics, along with logic; and set theory is now largely regarded as belonging to mathematics. Is Mathematics Invented or Discovered? The square root of two is irrational. And even when inventing, say, the internal combustion engine, the laws of physics which allowed such a device to exist were in place before the invention, and thus the particular arrangement of parts which effects its existence was discovered. We shouldn't expect more precision than is plausibly obtainable in a given field of study. Read this article and get some ideas about the theories that exist on this question.
And obviously the other one believes that mathematics is invented. Then, by using this i in your answer, any root can be expressed. I hold the same position for mathematic, it is a convention but not an arbitrary one. We never know whether the so called proof is correct. If mathematics would have built itself in isolation from the engineers and scientists, the chance for it to be remotly usefull would have been null.
So it is not always simple truth and beauty. This all seems to be a mirror of that old philosophical puzzle - if a tree falls in a forest and there is nobody there, does it make a sound? This naturally leads us to ask the question why mathematics is so effective at describing our universe — a question asked many times before by a number of great minds. I mean, it's a bit like asking wether a tree falling really makes a sound if nobody's there to hear it. Notwithstanding, some of it can be mitigated by way of the distinction between types and tokens, thereby, making it clear how software can be distinguished from hardware. Population genetics is splendid, but creativity has sort of gotten lost in all of this. Humans successfully able to find 118 elements in Periodic Table but they already exists.
But it is only formally so. Most believe discovered: Hyperarithmetic heirarchy The hyperarithmetic hierarchy is often phrased in terms of second order arithmetic, but I prefer to state it computationally. As argued above, it seems odd to think of natural languages as invented. And new ideas will always be fought. And this should be grounds for some pause. I have found things so magnificent that I was astounded….
In fact, I am not convinced that Euclid himself was entirely happy with his fifth postulate; in his book The Elements the proofs for his first 28 postulates do not make use of it. If they come from a special universe for wrong ideas, then discerning the difference is the same thing as inventing them. Monday, November 4, 2013 20% 100% To. Now let's say that human intelligence is the only cosmic intelligence and that human intellect has therefore invented mathematics since there is no even higher intelligence that has infused the cosmos with mathematics waiting to be discovered. Even though natural selection could explain why we cope with physical phenomena on the human scale, it could not explain by mathematics successfully deals with all scales- from atoms to galaxies. That's why we can find physical instances of pure mathematical objects.
According to that theory, atoms were tightly knotted tubes of ether- a mysterious substance which has now also been shown to not exist. The contradictory concept of married bachelors hadn't yet reared its head. In fact, I am not convinced that Euclid himself was entirely happy with his fifth postulate; in his book The Elements the proofs for his first 28 postulates do not make use of it. We can perhaps imagine a fully developed and consistent mathematical system which, in fact, would find no use. The problem is, is invention discovery or creation? Another example of this is imaginary numbers; the square root of minus one.