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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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I guess the main point I'm making is probably that the line between discovery and invention is not so neat as the question implies.
Once mathematicians have decided on the axioms they use (invention) then where the maths takes them is rather fixed, if not always expected (discovery). That probably means that maths is a glorious fusion of the two processes, working in tandem.
Once mathematicians have decided on the axioms they use (invention) then where the maths takes them is rather fixed, if not always expected (discovery). That probably means that maths is a glorious fusion of the two processes, working in tandem.
I love this series. Missed it last night, as I am travelling, but in a lifetime of thinking about the philosophy of science and maths, I have never asked this question, until Prof Fry raised it.
I also think she is very good at explaining some quite difficult concepts..
I still don't have an answer to her question, but I think mathematical analysis is essentially a tool (or series of tools) that can be used to model reality.
We can model five apples and three apples. The mathematical expression 5+3=8 helps us to predict that adding a group of five apples to a group of three apples will result in 8 apples.
That does not mean (in my view) that the cardinal numbers 3, 5 and 8 exist within the groups of apples. In my view, the numbers are a human construct used to model the apples.
In that case, the model is very accurate. In fact so accurate that it becomes very easy to confuse the model with the reality.
Newton developed some laws of motion
v=u+at
S=ut+ (at^2)/2
v^2=u^2 +2as
and so on.
This model can be used to make good predictions about the behaviour of a mass as it moves in a space- but only within certain constraints, such as ignoring air resistance.
As soon as we get into the real world, we need a more sophisticated model that takes into account the viscosity of the medium through which an object travels - colloquially, air resistance.
These models can be very accurate and indeed often we find that some remarkable insights fall out of them.
Prof. Fry gave the example of Maxwell's equations that model the behaviour of electric fields and magnetic fields in terms of conductivity and permittivity.
Remarkably, as Prof Fry showed, they can be used to predict that those waves propagate through free space at around 300,000 km/second, which is the speed of light. This gave the remarkable insight that light is an electromagnetic wave.
Bu they are still models that seek to represent reality.
Anyone who builds computer models or other models that mimic reality knows that here are certain constraints on those models. Once you try to apply the model outside those constraints, then the predictions coming out of the model rapidly become less reliable.
So I conclude that maths is an entirely human construct.
It is possible that other models can also be used to describe reality. Today most of western culture is taught that the mathematical models developed over 6000 years of scientific understanding are the only models that can succesfully represent the universe.
They are certainly the most successful, and have offered some deep insights. However, I suspect that they are not the only possible models.
I suspect, for example, that if we wish to understand consciousness, emotions and suchlike, then our numbers and calculus will not take us very far. Maybe a different type of model is needed.
I also think she is very good at explaining some quite difficult concepts..
I still don't have an answer to her question, but I think mathematical analysis is essentially a tool (or series of tools) that can be used to model reality.
We can model five apples and three apples. The mathematical expression 5+3=8 helps us to predict that adding a group of five apples to a group of three apples will result in 8 apples.
That does not mean (in my view) that the cardinal numbers 3, 5 and 8 exist within the groups of apples. In my view, the numbers are a human construct used to model the apples.
In that case, the model is very accurate. In fact so accurate that it becomes very easy to confuse the model with the reality.
Newton developed some laws of motion
v=u+at
S=ut+ (at^2)/2
v^2=u^2 +2as
and so on.
This model can be used to make good predictions about the behaviour of a mass as it moves in a space- but only within certain constraints, such as ignoring air resistance.
As soon as we get into the real world, we need a more sophisticated model that takes into account the viscosity of the medium through which an object travels - colloquially, air resistance.
These models can be very accurate and indeed often we find that some remarkable insights fall out of them.
Prof. Fry gave the example of Maxwell's equations that model the behaviour of electric fields and magnetic fields in terms of conductivity and permittivity.
Remarkably, as Prof Fry showed, they can be used to predict that those waves propagate through free space at around 300,000 km/second, which is the speed of light. This gave the remarkable insight that light is an electromagnetic wave.
Bu they are still models that seek to represent reality.
Anyone who builds computer models or other models that mimic reality knows that here are certain constraints on those models. Once you try to apply the model outside those constraints, then the predictions coming out of the model rapidly become less reliable.
So I conclude that maths is an entirely human construct.
It is possible that other models can also be used to describe reality. Today most of western culture is taught that the mathematical models developed over 6000 years of scientific understanding are the only models that can succesfully represent the universe.
They are certainly the most successful, and have offered some deep insights. However, I suspect that they are not the only possible models.
I suspect, for example, that if we wish to understand consciousness, emotions and suchlike, then our numbers and calculus will not take us very far. Maybe a different type of model is needed.
TTT:
2+2=4 is a universal truth
No. 2+2=4 is a self-consistent mathematical expression.
Two apples added to two apples wil result in four apples. That is what the mathematical expression you quoted above predicts.
However, when we add two apples to two pears, we do not get four apples. The assumptions behind the model have not been adequately defined - we are going outside the model, so the predictions become unreliable.
Beore we can use the expression 2+2=4 to make meaningful predictions, we have to we have to define the limits of the model.
In making that definition, we are humans constructing a model.
It's OK, if you don't see it that way. It's just something I thought up while listening to a lecture about electric vehicles and sensorisation...
2+2=4 is a universal truth
No. 2+2=4 is a self-consistent mathematical expression.
Two apples added to two apples wil result in four apples. That is what the mathematical expression you quoted above predicts.
However, when we add two apples to two pears, we do not get four apples. The assumptions behind the model have not been adequately defined - we are going outside the model, so the predictions become unreliable.
Beore we can use the expression 2+2=4 to make meaningful predictions, we have to we have to define the limits of the model.
In making that definition, we are humans constructing a model.
It's OK, if you don't see it that way. It's just something I thought up while listening to a lecture about electric vehicles and sensorisation...
ijk and Jim
//but in a lifetime of thinking about the philosophy of science and maths, I hve never thought about it// ijk
well you dont have to - some one has done it before
so all you have to do is read
Plato(*) - ideas have a separate existence
and here is Brouwer(*) again
https:/
'is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles'
and all a maths grad can say above is
"Brouwer the hoova - who he den ? foo! " I remain deeply shocked
where were you two when these lectures were being given on your maff courses? - darn the boozer quaffin halves of lager and ordering pink gins and discussing footie?
and no one but me knows that Euler(*) wrote - "notione non notatione" - by notion/idea not by notation. Clearly platonist - that is ideas have a separate existence
(*) who all dey den - maff or wha' ?
//but in a lifetime of thinking about the philosophy of science and maths, I hve never thought about it// ijk
well you dont have to - some one has done it before
so all you have to do is read
Plato(*) - ideas have a separate existence
and here is Brouwer(*) again
https:/
'is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles'
and all a maths grad can say above is
"Brouwer the hoova - who he den ? foo! " I remain deeply shocked
where were you two when these lectures were being given on your maff courses? - darn the boozer quaffin halves of lager and ordering pink gins and discussing footie?
and no one but me knows that Euler(*) wrote - "notione non notatione" - by notion/idea not by notation. Clearly platonist - that is ideas have a separate existence
(*) who all dey den - maff or wha' ?
True, but all of those are different notational ways of expressing the same result, which is that .. and .. put together make ....
I would still hold that, as usual, it's neither useful nor very sensible to insist that there is a strict choice between maths being discovered and invented. Once mathematicians have decided on the rules, then certain truths are forced upon them, but not all of the possible rules have obvious physical motivations, and sometimes mathematicians genuinely are playing around, in an idealised universe that has no bearing to anything in the real world. If so, is that not an invention?
I would still hold that, as usual, it's neither useful nor very sensible to insist that there is a strict choice between maths being discovered and invented. Once mathematicians have decided on the rules, then certain truths are forced upon them, but not all of the possible rules have obvious physical motivations, and sometimes mathematicians genuinely are playing around, in an idealised universe that has no bearing to anything in the real world. If so, is that not an invention?
>>> and sometimes mathematicians genuinely are playing around, in an idealised universe that has no bearing to anything in the real world. If so, is that not an invention?
Agreed. n-dimensional space, where n doesn't even have to be an integer or even rational, can be great fun to play around with in one's mind and on paper but, since it doesn't actually 'exist', it's hard to see how any part of it can ever have been 'discovered'.
Agreed. n-dimensional space, where n doesn't even have to be an integer or even rational, can be great fun to play around with in one's mind and on paper but, since it doesn't actually 'exist', it's hard to see how any part of it can ever have been 'discovered'.
// n-dimensional space, where n isnt an integer, can be great fun to play around with in one's mind and on paper but, since it doesn't actually 'exist',//
er sorry the fractals - serenyi sponge and the gomoboola triangle do not have whole number dimentions the triangle thingey is 1.58
you knew I was gonna do this didnt you ?
er sorry the fractals - serenyi sponge and the gomoboola triangle do not have whole number dimentions the triangle thingey is 1.58
you knew I was gonna do this didnt you ?
// 2 apples and 2 kangaroos don't make anyfin'//
sure they do - four fings !
and in the prog - Prof Fry's little frenz - al the sensible stuff they said is on the cutting room floor - and when they start waving their hands and saying - "its bigger than you think!" eyes bulging
and Prof Fry demurely says - I often hear that !
that DOES get into the prog
sure they do - four fings !
and in the prog - Prof Fry's little frenz - al the sensible stuff they said is on the cutting room floor - and when they start waving their hands and saying - "its bigger than you think!" eyes bulging
and Prof Fry demurely says - I often hear that !
that DOES get into the prog
god created adam from dust & eve from adam's rib. 1 rib! Beginnings of maths = discovery by god & start of humanity ;)
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