Pythagoras' Theorem. Evidence of elsewhere or our ingenuity.

[JimC]

Jim, how do you factor in gas giants in your definition of the discreteness of planethood? It is possible to consider them as nothing but atmosphere! Just asking...

Yes, their discontinuity is different (and somewhat less abrupt) than that exhibited by a rocky planet with an atmosphere. However, there is still a relatively sudden change in density, as any photo of Jupiter or Saturn will show - they do not appear as misty objects, with a smooth steady change in opacity...

[JimC]

I think most of my disdain for Plato's Forms (which seems to be seeping out in this thread) is that I'm always skeptical of notions of perfection. Is the circle with all the points equidistant from the center the best circle, or merely the most simple and generic one conceivable?

It's worse than you might think. To truly conceive of a perfect circle, sphere, line, right-angle, etc., you must first conceive the idea of a perfectly stable, flat portion of the universe and that is only possible if there is no matter whatsoever anywhere in that universe!

Have fun.

This implies that perfectionism is a more dangerous tendency that previously believed.

And one way in which maths differs from empirical science. Maths starts from perfection and struggles to apply itself to the world. Science works from the other end.

Perhaps it is the cognitive dissonance between these two approaches to understanding that leads to the most fruitful areas of exploration. When comparing observational evidence with predictions based on a mathematical model, the discrepancies are valuable pointers to refining our understanding...

[Xamonas Chegwé]

I think most of my disdain for Plato's Forms (which seems to be seeping out in this thread) is that I'm always skeptical of notions of perfection. Is the circle with all the points equidistant from the center the best circle, or merely the most simple and generic one conceivable?

It's worse than you might think. To truly conceive of a perfect circle, sphere, line, right-angle, etc., you must first conceive the idea of a perfectly stable, flat portion of the universe and that is only possible if there is no matter whatsoever anywhere in that universe!

Have fun.

This implies that perfectionism is a more dangerous tendency that previously believed.

And one way in which maths differs from empirical science. Maths starts from perfection and struggles to apply itself to the world. Science works from the other end.

Perhaps it is the cognitive dissonance between these two approaches to understanding that leads to the most fruitful areas of exploration. When comparing observational evidence with predictions based on a mathematical model, the discrepancies are valuable pointers to refining our understanding...

Absolutely!

The history of scientific progress is basically a history of improved measuring accuracy throwing up discrepancies in previous models and showing the need for refinement. That's where we naked monkeys reach for the sliderule again...

[Hermit]

Does this define something real or have we just used clever symbology to create a context in which it is always true?

Maybe both?

I regard Pythagoras' theorem as a particular instance of mathematics in general. It's borne of our observations of the world around us, and it works.

If mathematical abstractions did not define something real, the theory of gravity would not have set astronomers on a search of a necessary body to explain away some incongruities of the planetary orbits within our solar system. After several decades of looking for that necessary body, Pluto was discovered. No apophenia there, nor revelation.

This does not imply that mathematical theorems are not at all clever concepts in which they are always true. The theory of relativity proved that the theory of gravity is no more than a fair approximation of what we have conceptualised the real world to be, and the latter has its own limitations. And then came quantum theory...

So, I think that Pythagoras' theorem in particular - and mathematical theorems in general - are a product of observations we make regarding reality as well as a fabrication, and I loathe further metaphysical considerations about it. In matters mathematical we don't really need to believe in the ontological sphere of existence and we need not insist on ever getting to the bottom of it all - ultimate reality. It is enough to realise that mathematical theorems are borne of observation and experience and lead to further discoveries and uses in the real world.

[pErvinalia]

I second that last paragraph. When the metaphysical wibble starts about mathematics, I turn right off. Some people want mathematics to replace God. Can't we just be satisfied with the beauty of our garden without imagining fairies at the bottom of it too?

[JimC]

Hermit wrote:

So, I think that Pythagoras' theorem in particular - and mathematical theorems in general - are a product of observations we make regarding reality as well as a fabrication, and I loathe further metaphysical considerations about it. In matters mathematical we don't really need to believe in the ontological sphere of existence and we need not insist on ever getting to the bottom of it all - ultimate reality. It is enough to realise that mathematical theorems are borne of observation and experience and lead to further discoveries and uses in the real world.

Fabrication is a word that may have been chosen with malice aforethought.

It is not incorrect, but has clear implications of lies.

Much of modern mathematics is a construction, a development of patterns inherent in earlier discoveries about logical relationships. Parts of the flowering of this rigorous intellectual endeavour have found reflections in our exploration of the physical universe, parts are simply a beautiful flower of logic, sailing in the void of possible realities, to be admired by those with the wit to do so...

Which is precious few...

[Audley Strange]

I second that last paragraph. When the metaphysical wibble starts about mathematics, I turn right off. Some people want mathematics to replace God. Can't we just be satisfied with the beauty of our garden without imagining fairies at the bottom of it too?

Well that's what started my thinking of it. To me Platonic forms do seem like fairies at the bottom of the garden. Problem is they seemed to be there and I wanted to get my thinking straight on this.

[JimC]

I second that last paragraph. When the metaphysical wibble starts about mathematics, I turn right off. Some people want mathematics to replace God. Can't we just be satisfied with the beauty of our garden without imagining fairies at the bottom of it too?

Well that's what started my thinking of it. To me Platonic forms do seem like fairies at the bottom of the garden. Problem is they seemed to be there and I wanted to get my thinking straight on this.

There is a considerable difference. For some, fairies were a real part of the physical universe, not a metaphysical construct.

Platonic forms, in the lightest sense of the term, were simply abstractions of idealised mathematical objects, existing in the universe of human thought, and none the worse for that...

Somewhat more mystical mathematicians gave them a quasi-real existence in a meta-universe. Not really needed, IMO...

[Audley Strange]

Well yes, the problem is I can't shake a specific type of mysticism as a really useful metaphoric tool for certain things and sometimes it leaks across and that was one of these times where I was considering some kind of functional reality to plane geometry inherent in the universe itself rather than being an elaborate measuring tool. I get it. Centimetres and Feet and Cubits are all measurements, all manmade and specific. They are useful for negotiation through territory but are not inherent in it.

[JimC]

Audley Strange wrote:

...functional reality to plane geometry inherent in the universe itself rather than being an elaborate measuring tool...

There is no inherent barrier to there being a geometry inherent to the universe. Perhaps one of the many that can be constructed via the hominid mind...

Oddly, it may be a geometry that changes as a function of the age of the universe, but that is not a disqualification...

[Hermit]

So, I think that Pythagoras' theorem in particular - and mathematical theorems in general - are a product of observations we make regarding reality as well as a fabrication, and I loathe further metaphysical considerations about it. In matters mathematical we don't really need to believe in the ontological sphere of existence and we need not insist on ever getting to the bottom of it all - ultimate reality. It is enough to realise that mathematical theorems are borne of observation and experience and lead to further discoveries and uses in the real world.

Fabrication is a word that may have been chosen with malice aforethought.

It is not incorrect, but has clear implications of lies.

Much of modern mathematics is a construction, a development of patterns inherent in earlier discoveries about logical relationships. Parts of the flowering of this rigorous intellectual endeavour have found reflections in our exploration of the physical universe, parts are simply a beautiful flower of logic, sailing in the void of possible realities, to be admired by those with the wit to do so...

Which is precious few...

"Fabrication" was not used in any malicious way. Think of it as "abstractions (or conceptualisations) of what we have perceived in the physical world." Even though there is no such thing as a perfect circle in the real world, we have fabricated a concept of "circle" from perhaps looking at the sun or moon, then gone on to fabricate radius, diameter, pi and so on, and those concepts work marvellously. Quite obviously, to go on to another example, we know fully well that no two apples are exactly alike, we developed a concepts of discreteness, uniformity, addition, multiplication and so on. We treated objects as if they were exactly the same even though they turned out to be not quite so, but that did not stop mathematics and science from being monumentally useful.

So, "fabrication" was not meant as a put-down. It was meant to tease out the relationship between reality and what we make of it. It was meant to elucidate why I think that we need neither insist on an ontological sphere of existence of mathematical elements nor being able to ever be able to even be in a position to know "ultimate reality." In short, we fabricate or abstract idealised concepts from our experience, and as the proverb goes, the (provisional) proof of the pudding is in the eating."

[JimC]

So, I think that Pythagoras' theorem in particular - and mathematical theorems in general - are a product of observations we make regarding reality as well as a fabrication, and I loathe further metaphysical considerations about it. In matters mathematical we don't really need to believe in the ontological sphere of existence and we need not insist on ever getting to the bottom of it all - ultimate reality. It is enough to realise that mathematical theorems are borne of observation and experience and lead to further discoveries and uses in the real world.

Fabrication is a word that may have been chosen with malice aforethought.

It is not incorrect, but has clear implications of lies.

Much of modern mathematics is a construction, a development of patterns inherent in earlier discoveries about logical relationships. Parts of the flowering of this rigorous intellectual endeavour have found reflections in our exploration of the physical universe, parts are simply a beautiful flower of logic, sailing in the void of possible realities, to be admired by those with the wit to do so...

Which is precious few...

"Fabrication" was not used in any malicious way. Think of it as "abstractions (or conceptualisations) of what we have perceived in the physical world." Even though there is no such thing as a perfect circle in the real world, we have fabricated a concept of "circle" from perhaps looking at the sun or moon, then gone on to fabricate radius, diameter, pi and so on, and those concepts work marvellously. Quite obviously, to go on to another example, we know fully well that no two apples are exactly alike, we developed a concepts of discreteness, uniformity, addition, multiplication and so on. We treated objects as if they were exactly the same even though they turned out to be not quite so, but that did not stop mathematics and science from being monumentally useful.


So, "fabrication" was not meant as a put-down. It was meant to tease out the relationship between reality and what we make of it. It was meant to elucidate why I think that we need neither insist on an ontological sphere of existence of mathematical elements nor being able to ever be able to even be in a position to know "ultimate reality." In short, we fabricate or abstract idealised concepts from our experiences.