Connections # 323- Mon 29 April
Quizzes & Puzzles1 min ago
Is Mathematics There To Be Discovered Or Invented?
84 Answers
This is a question that I have pondered for some time.
Which numbers can be found in nature?
I raised a question two months ago asking -
Who invented binary numbers?
This led to the answers that I wanted and also to an interesting thread about the nature or lack of it of mathematics. So, I thought that I would put forward this question. If you are interested in the idea I would encourage you to look at the thread about Who invented binary numbers? which should not be too difficult to find. But please, do not continue that discussion on that thread. Start the new discussion here. Thanks.
So, this thread should be about whether mathematics exists and is to be discovered or whether it is an invention. If some mathematics exists and some must be invented then the thread may be about what exists mathematically and where can it be discovered.
And before we begin...
I hope that you enjoy this.
Which numbers can be found in nature?
I raised a question two months ago asking -
Who invented binary numbers?
This led to the answers that I wanted and also to an interesting thread about the nature or lack of it of mathematics. So, I thought that I would put forward this question. If you are interested in the idea I would encourage you to look at the thread about Who invented binary numbers? which should not be too difficult to find. But please, do not continue that discussion on that thread. Start the new discussion here. Thanks.
So, this thread should be about whether mathematics exists and is to be discovered or whether it is an invention. If some mathematics exists and some must be invented then the thread may be about what exists mathematically and where can it be discovered.
And before we begin...
I hope that you enjoy this.
Answers
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Jim, that link is typical of the media not understanding what they are reporting. Plant behaviour is mostly governed by hormones and the relationships between the effects of various hormones produce the end result. These can be described mathematically, but the plants don't need the maths, it is the scientist that need the maths but that is only as an aid to understanding, the maths is not an end in itself.
I don't know that I agree that just because we find complex numbers convenient to solve problems from QM to third order polynomials that makes them somehow 'intrinsic' or discovered.
They are a construct which can be used to solve problems - much in the way a negative number can be used to represent a debt.
We define the behaviour of a negative number - for example that 'multiplying' two together gives a positive
We define the behaviour of complex mumbers - for example that dividing is multiplying a complex number by its conjugate
These definitions give us answers that work for us
if they did not work we would seek other mathematics that did work.
Maths is as Marcus Du Sautoy likes to say all about patterns
It is these patterns that are fundamental to nature - maths is a formalisation of techniques and representations that allow us to identify these patterns, predict them and represent them.
So when complex numbers were invented it was as a tool that solved a problem (finding the roots of third order polynomials) what then happened was this tool gave us ways to represent other objects (like elctronic circuits or quantum mechanics) - this gave us a deeper pattern and showed us a connection between these things where we had not previously expected one.
Sometimes these connections are so unexpected that we are shocked by them into an almost mystical sense of wonder
Like Eulers' identity that e to the power i pi is minus 1
http://
these are the connections between the patterns that we discover
They are a construct which can be used to solve problems - much in the way a negative number can be used to represent a debt.
We define the behaviour of a negative number - for example that 'multiplying' two together gives a positive
We define the behaviour of complex mumbers - for example that dividing is multiplying a complex number by its conjugate
These definitions give us answers that work for us
if they did not work we would seek other mathematics that did work.
Maths is as Marcus Du Sautoy likes to say all about patterns
It is these patterns that are fundamental to nature - maths is a formalisation of techniques and representations that allow us to identify these patterns, predict them and represent them.
So when complex numbers were invented it was as a tool that solved a problem (finding the roots of third order polynomials) what then happened was this tool gave us ways to represent other objects (like elctronic circuits or quantum mechanics) - this gave us a deeper pattern and showed us a connection between these things where we had not previously expected one.
Sometimes these connections are so unexpected that we are shocked by them into an almost mystical sense of wonder
Like Eulers' identity that e to the power i pi is minus 1
http://
these are the connections between the patterns that we discover
Ah, the cubic polynomials problem! I think that is a case of something being initially discovered in maths. A lot of this comes down to interpretation, of course. But in terms of timing, i or root -1 first appeared in a calculation before someone had thought to use it. Then again, the method of that calculation can probably be called an invention.
I think that's the usual way in maths. Once you invent a new technique, or new field, or a new equation, you often have little or no idea where that will take you. And it may not take you anywhere, if it comes to that.
I think that's the usual way in maths. Once you invent a new technique, or new field, or a new equation, you often have little or no idea where that will take you. And it may not take you anywhere, if it comes to that.
@ANYBODY who read Methyl's quote about gibberish.
Is it really not understandable. I will happily rewrite it if others cannot understand. Otherwise I'll let it lie.
@ALL
It appears that numbers are of interest here. What about operators?
@bert_h
Thanks, I like this explanation. It reduces a squaring operation to incremental addition. The next step is to consider whether this is actually an example of nature adding anything. I would be very happy to find some clear examples of operations happening to numbers in nature.
Is it really not understandable. I will happily rewrite it if others cannot understand. Otherwise I'll let it lie.
@ALL
It appears that numbers are of interest here. What about operators?
@bert_h
Thanks, I like this explanation. It reduces a squaring operation to incremental addition. The next step is to consider whether this is actually an example of nature adding anything. I would be very happy to find some clear examples of operations happening to numbers in nature.
@jim
in an earlier reply I gave a reference to a blog where a similar discussion to this one is recorded. I am not basing my views on this, it's just that I found it when thinking how to reply to @Methyl, and thought it would save me a lot of time to link to it rather than give a long explanation here.
Here it is again http://
Some easier to understand examples of complex numbers being essential are:
- Bombelli's work on cubic equations which established that real (in the mathematical sense) problems require complex (in the mathematical sense) arithmetic for their solution.
- another example occurs when looking at power series for 1/(1-x^2) and for 1/(1+x^2). It is found that the first converges if and only if -1
in an earlier reply I gave a reference to a blog where a similar discussion to this one is recorded. I am not basing my views on this, it's just that I found it when thinking how to reply to @Methyl, and thought it would save me a lot of time to link to it rather than give a long explanation here.
Here it is again http://
Some easier to understand examples of complex numbers being essential are:
- Bombelli's work on cubic equations which established that real (in the mathematical sense) problems require complex (in the mathematical sense) arithmetic for their solution.
- another example occurs when looking at power series for 1/(1-x^2) and for 1/(1+x^2). It is found that the first converges if and only if -1
"Can anyone find examples in nature where mathematical operations are performed."
Most of you have lost me! But does the sahara desert ant not carry out some method of mathematical performance? It appears to measure the angle of the sun and work out how much time it can spend scurrying about, including counting its steps from its hidey hole and retrackes them back, before it frazzles.
Probably off tangent, but I found it interesting.
Most of you have lost me! But does the sahara desert ant not carry out some method of mathematical performance? It appears to measure the angle of the sun and work out how much time it can spend scurrying about, including counting its steps from its hidey hole and retrackes them back, before it frazzles.
Probably off tangent, but I found it interesting.
@jim (posting again since some of my post didn't appear for some reason. I think its because of the 'less than' sign, which I have avoided this time)
in an earlier reply I gave a reference to a blog where a similar discussion to this one is recorded. I am not basing my views on this, it's just that I found it when thinking how to reply to @Methyl, and thought it would save me a lot of time to link to it rather than give a long explanation here.
Here it is again http://
Some easier to understand examples of complex numbers being essential are:
- Bombelli's work on cubic equations which established that real (in the mathematical sense) problems require complex (in the mathematical sense) arithmetic for their solution.
- another example occurs when looking at power series about the origin for 1/(1-x^2) and for 1/(1+x^2). It is found that the first converges if and only if -1 less than x less than 1 and this makes sense since the function has singularities at x=-1 and x=+1.
However the power series for the second function also only converges in the interval -1 less than x less than 1, which seems strange since the function has no singularities along the real axis.
However this mystery is solved if we go to the complex plane, since we find singularities at x=-i and x=+i and so both functions have series which converge within the unit circle in the complex plane, since the singularities are all at distance 1 from the origin.
in an earlier reply I gave a reference to a blog where a similar discussion to this one is recorded. I am not basing my views on this, it's just that I found it when thinking how to reply to @Methyl, and thought it would save me a lot of time to link to it rather than give a long explanation here.
Here it is again http://
Some easier to understand examples of complex numbers being essential are:
- Bombelli's work on cubic equations which established that real (in the mathematical sense) problems require complex (in the mathematical sense) arithmetic for their solution.
- another example occurs when looking at power series about the origin for 1/(1-x^2) and for 1/(1+x^2). It is found that the first converges if and only if -1 less than x less than 1 and this makes sense since the function has singularities at x=-1 and x=+1.
However the power series for the second function also only converges in the interval -1 less than x less than 1, which seems strange since the function has no singularities along the real axis.
However this mystery is solved if we go to the complex plane, since we find singularities at x=-i and x=+i and so both functions have series which converge within the unit circle in the complex plane, since the singularities are all at distance 1 from the origin.
essential - for what? vascop solutions?
You're talking about complex planes and convergences - are these all not just Human constructs?
The key test to whether something is invented or discovered is whether it would exist if humanity did not.
Did the complex plane exist before humanity? I find that hard to see
You're talking about complex planes and convergences - are these all not just Human constructs?
The key test to whether something is invented or discovered is whether it would exist if humanity did not.
Did the complex plane exist before humanity? I find that hard to see
@jake
Try reading 'The Emperor's New mind' by Roger Penrose, Oxford Press, especially Chapter 3 Mathematics and Reality.
He talks about the Mandelbrot set and says, I quote:"..the complete details of the complications of the structure of Mandelbrot's set cannot really be fully comprehended by any one of us, nor can it be fully revealed by any computer. It would seem that this structure is not just part of our minds, but it has a reality of its own....The Mandelbrot set is not an invention of the human mind: it was a discovery. Like Mount Everest, the Mandelbrot set is just there!"
To explain the rule for generating the Mandelbrot set we need complex number theory.
Try reading 'The Emperor's New mind' by Roger Penrose, Oxford Press, especially Chapter 3 Mathematics and Reality.
He talks about the Mandelbrot set and says, I quote:"..the complete details of the complications of the structure of Mandelbrot's set cannot really be fully comprehended by any one of us, nor can it be fully revealed by any computer. It would seem that this structure is not just part of our minds, but it has a reality of its own....The Mandelbrot set is not an invention of the human mind: it was a discovery. Like Mount Everest, the Mandelbrot set is just there!"
To explain the rule for generating the Mandelbrot set we need complex number theory.
@justnotcricket
If you are interested in mathematics in nature, I'd suggest reading:
Mathematics of Life: Unlocking the Secrets of Existence by Ian Stewart - Emeritus Professor of Mathematics at Warwick University.
It shows how mathematics is interacting with biology more and more, and is one of the hottest areas of science.
If you are interested in mathematics in nature, I'd suggest reading:
Mathematics of Life: Unlocking the Secrets of Existence by Ian Stewart - Emeritus Professor of Mathematics at Warwick University.
It shows how mathematics is interacting with biology more and more, and is one of the hottest areas of science.
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