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Maths: Discovered Or Invented?
”Magic Numbers: Hannah Fry’s Mysterious World of Maths”, a BBC Four documentary series in which Dr Hannah Fry explores the mystery of maths and asks “Is maths invented like a language or is it discovered and part of the fabric of the universe?”
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yeah apparently
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I was gonna say - I can't imagine! but I am not sure if some here would understand
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I was gonna say - I can't imagine! but I am not sure if some here would understand
In my opinion, considering everything obeys mathematical laws, even the most fundamental particles, or rather quantum fields, the laws of mathematics, are in fact, to call it by another name, "information."
So, what is the source of information?
We know from our own day to day experience, that it requires a, "mind," to conceive information and provide the impetus for action.
A computer is a useless piece of junk until a programmer has input information as a programme.
So information, ie mathematics, has to originate in a mind.
It cannot evolve.
That is why I believe that the origin of the information to create and maintain our universe, is an intelligent mind.
In my opinion.
So, what is the source of information?
We know from our own day to day experience, that it requires a, "mind," to conceive information and provide the impetus for action.
A computer is a useless piece of junk until a programmer has input information as a programme.
So information, ie mathematics, has to originate in a mind.
It cannot evolve.
That is why I believe that the origin of the information to create and maintain our universe, is an intelligent mind.
In my opinion.
The story of the square root of -1 is one of my favourite parts of mathematical history. And it's definitely a discovery, one that confused mathematicians for easily a couple of centuries before it was properly understood.
Long story short, it turns up in the theory of cubic equations -- sometimes the solutions are nice real numbers but must be expressed in terms of the square root of minus one and there's no way around this - called the "irreducible case". Eventually this annoyance became essential for maths, but there's no doubting that it was an accidental -- and profound! -- discovery.
Long story short, it turns up in the theory of cubic equations -- sometimes the solutions are nice real numbers but must be expressed in terms of the square root of minus one and there's no way around this - called the "irreducible case". Eventually this annoyance became essential for maths, but there's no doubting that it was an accidental -- and profound! -- discovery.
root (-1) is very useful, but stil is part of the modelling process.
IN maths it's a means to solve equations that donl't have a real solution.
In engineering, it is a way of representing a 2-dimensional field (with x- and y- axes) using a single equation; the real numbers (1, 2, 3, 4 etc) representing the x-axis and the imaginary numbers (i, 2i, 3i etc) representing the y-axis.
It's called an Argand diagram, for those who are interested.
However, I stil think these things are human constructs, created in our efforts to better understand an analytic and reductioninst universe, rather than inherent in nature, which is not necessarily reductionist.
Ad to PP - half the fun of these is to think for oneself. I may well be wrong, but bullsh1tting about obscure philosophical issues after a wonderful evening in a German Bierkeller in Dresden is so much more fun than reading Luitzen Egbertus Jan Brouwer's indigestible tracts.
And by the way, I never studied maths at degree. Only engineering. Brouwer and the like were way too theoretical and obscure for the poor, non-numerate engineers. We just did the partial differentials and the conformal mappings without trying to understand any of the wonderful theory behind it all.
It's only in later life that I discovered that stuff.
IN maths it's a means to solve equations that donl't have a real solution.
In engineering, it is a way of representing a 2-dimensional field (with x- and y- axes) using a single equation; the real numbers (1, 2, 3, 4 etc) representing the x-axis and the imaginary numbers (i, 2i, 3i etc) representing the y-axis.
It's called an Argand diagram, for those who are interested.
However, I stil think these things are human constructs, created in our efforts to better understand an analytic and reductioninst universe, rather than inherent in nature, which is not necessarily reductionist.
Ad to PP - half the fun of these is to think for oneself. I may well be wrong, but bullsh1tting about obscure philosophical issues after a wonderful evening in a German Bierkeller in Dresden is so much more fun than reading Luitzen Egbertus Jan Brouwer's indigestible tracts.
And by the way, I never studied maths at degree. Only engineering. Brouwer and the like were way too theoretical and obscure for the poor, non-numerate engineers. We just did the partial differentials and the conformal mappings without trying to understand any of the wonderful theory behind it all.
It's only in later life that I discovered that stuff.
IJKLM -- the reason I'm going to regard i = sqrt(-1) as a special case is because its history simply doesn't support the idea that it was invented. It's well worth a read, and very enlightening, to study the history of that number through the 15th-17th centuries, when it was first discovered and investigated.
See also my earlier post -- to that I should perhaps add that, at the same time as sqrt(-1) was first encountered, the only solutions that any mathematician accepted as "real" were those in positive numbers. Descartes wrote in his textbook "La Géométrie", among other things, that "sometimes it happens that some of the solutions [to a quadratic equation] are false, or less than nothing". It stands to reason that if people don't accept the reality of negative numbers then there's no reason to invent their square roots either. What's the square root of meaningless bull? A meaningless bullock, perhaps.
Instead, as I say, the study of cubic equations led to the remarkable result that some equations whose solutions were definitely positive whole numbers could be expressed in terms of this bizarre, and completely meaningless, object called sqrt(-1). What nonsense! And yet there it is, staring you in the face, forcing you to accept its existence. There is no other word for it: this was a discovery. Nobody asked for it, nor did they demand it; the fact that this strange object turns out to be so useful is the purest of accidents.
See also my earlier post -- to that I should perhaps add that, at the same time as sqrt(-1) was first encountered, the only solutions that any mathematician accepted as "real" were those in positive numbers. Descartes wrote in his textbook "La Géométrie", among other things, that "sometimes it happens that some of the solutions [to a quadratic equation] are false, or less than nothing". It stands to reason that if people don't accept the reality of negative numbers then there's no reason to invent their square roots either. What's the square root of meaningless bull? A meaningless bullock, perhaps.
Instead, as I say, the study of cubic equations led to the remarkable result that some equations whose solutions were definitely positive whole numbers could be expressed in terms of this bizarre, and completely meaningless, object called sqrt(-1). What nonsense! And yet there it is, staring you in the face, forcing you to accept its existence. There is no other word for it: this was a discovery. Nobody asked for it, nor did they demand it; the fact that this strange object turns out to be so useful is the purest of accidents.
Guess it depends on what you mean by "making more sense". Babylonians enjoyed working with the number 60, so to them 360 would have been far more natural than 10, 100 or 1000 might be to us. I think the obsession with 10 as a true base, taken to its proper extreme, only really caught on as an indirect result of the French Revolution.
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