Assuming you can draw two lines of length precisely 1, then you can draw a line of length precisely sqrt(2), by Pythagoras, which is an irrational number. However, surely it is impossible to draw a line of length sqrt(2), as we live in a finite universe?

Where is the problem with this? Does the fact we live in curved space perhaps mean that Pythagoras' theorem isn't exactly correct (as, due to the curvature of space, the sum of the angles in a triangle isn't actually 180 degrees), or perhaps it's something to do with particles?

Just a small thought I had in church yesterday, that I thought interesting enough to share...

Originally posted by Swlabr Assuming you can draw two lines of length precisely 1, then you can draw a line of length precisely sqrt(2), by Pythagoras, which is an irrational number. However, surely it is impossible to draw a line of length sqrt(2), as we live in a finite universe?

Where is the problem with this? Does the fact we live in curved space perhaps mean that Pythagoras' the ...[text shortened]... ust a small thought I had in church yesterday, that I thought interesting enough to share...

Mathematics is the best of all worlds. Because it is invented. It is not the real world. More as an approximation, of course.

In the real world, nothing is exactly as in mathematics. Say a circle, as an example. Circles doesn not exist in the real world.

Originally posted by Swlabr Assuming you can draw two lines of length precisely 1, then you can draw a line of length precisely sqrt(2), by Pythagoras, which is an irrational number. However, surely it is impossible to draw a line of length sqrt(2), as we live in a finite universe?

Where is the problem with this? Does the fact we live in curved space perhaps mean that Pythagoras' the ...[text shortened]... ust a small thought I had in church yesterday, that I thought interesting enough to share...

Practicality aside, I think relativity, quantum jitters and the smeared-out nature of strings in string theory all preclude the possibility of being able to measure any constructed line with infinite accuracy.

A line is one-dimensional. Matter is (at least) 3-dimensional. You can draw lines only with matter and not with hypothetical one-dimensional constructs.

Originally posted by Swlabr Assuming you can draw two lines of length precisely 1, then you can draw a line of length precisely sqrt(2), by Pythagoras, which is an irrational number. However, surely it is impossible to draw a line of length sqrt(2), as we live in a finite universe?

Where is the problem with this? Does the fact we live in curved space perhaps mean that Pythagoras' the ...[text shortened]... ust a small thought I had in church yesterday, that I thought interesting enough to share...

I would say the problem is the assumption that numbers actually exist. Numbers are a human construct like language that we use to describe the world around us. So yes, a line exists and it is so long. But to say it is '1' or 'sqrt(2)' long is only our description of it.

The map is not the territory, one might say.

True, there is potentially less debate around the notion of 'four' compared to the notion of 'chicken'. But both are just human idea constructs -- not absolutes.

Originally posted by spruce112358 I would say the problem is the assumption that numbers actually exist. Numbers are a human construct like language that we use to describe the world around us. So yes, a line exists and it is so long. But to say it is '1' or 'sqrt(2)' long is only our description of it.

The map is not the territory, one might say.

True, there is potentially less d ...[text shortened]... ed to the notion of 'chicken'. But both are just human idea constructs -- not absolutes.

But surely the notion of "number" is not merely a human construction - there is still one chicken there, humans have merely put a name to this concept of one-ness. Mathematics is pure observation, but it is an observation that is dependent on a deeper concept than merely numbers. Mathematics still hold whether you count base 10, 60 or 1729. It all still works.

Further, surely the concept of number is also not dependent on our universe?

Originally posted by Swlabr But surely the notion of "number" is not merely a human construction - there is still one chicken there, humans have merely put a name to this concept of one-ness. Mathematics is pure observation, but it is an observation that is dependent on a deeper concept than merely numbers. Mathematics still hold whether you count base 10, 60 or 1729. It all still works ...[text shortened]... Further, surely the concept of number is also not dependent on our universe?

emm...discuss?

I think it may depend on the particular physical context:
There may physically exist EXACTLY two of something somewhere (EXACTLY two protons in a Helium nucleus etc) in physical reality but it is physically impossible to draw a line that is physically EXACTLY of length sqrt(2).

-so sometimes the concept of number EXACTLY corresponds to physical reality but sometimes it does NOT and is merely a convenient approximation.

Originally posted by FabianFnas Mathematics is the best of all worlds. Because it is invented. It is not the real world. More as an approximation, of course.

In the real world, nothing is exactly as in mathematics. Say a circle, as an example. Circles doesn not exist in the real world.

The marbles I had as a child existed as a glob of lots of circles, very tightly compressed and touching each other.

Originally posted by Andrew Hamilton I think it may depend on the particular physical context:
There may physically exist EXACTLY two of something somewhere (EXACTLY two protons in a Helium nucleus etc) in physical reality but it is physically impossible to draw a line that is physically EXACTLY of length sqrt(2).

-so sometimes the concept of number EXACTLY corresponds to physical reality but sometimes it does NOT and is merely a convenient approximation.

So, basically, you're saying there there are the cardinal numbers - numbers that hold information about the size of sets, 1,2,... - and that these are the only "true" numbers. Everything else is a hypothetical mathematical construction built around these and are used to measure things like lines, sums of money etc. ?

Originally posted by Swlabr So, basically, you're saying there there are the cardinal numbers - numbers that hold information about the size of sets, 1,2,... - and that these are the only "true" numbers. Everything else is a hypothetical mathematical construction built around these and are used to measure things like lines, sums of money etc. ?

If the criterion for defining a “true” number is that it must EXACTLY corresponds to physical reality, then I would say not just sometimes the cardinal numbers but sometimes the rational fractions (e.g. a helium atom has EXACTLY half the protons as a hydrogen atom) but not always because it depends on the context (I cannot pour out EXACTLY one litre of water). I assume that in most contexts (if not all) all other types of number do not EXACTLY corresponds to physical reality but I cannot help but wonder if there may be some physical context that we are currently unaware of where, for example, something can be an EXACT irrational number in magnitude? -or is that just always nonsense?

Originally posted by Andrew Hamilton If the criterion for defining a “true” number is that it must EXACTLY corresponds to physical reality, then I would say not just sometimes the cardinal numbers but sometimes the rational fractions (e.g. a helium atom has EXACTLY half the protons as a hydrogen atom) but not always because it depends on the context (I cannot pour out EXACTLY one litre ...[text shortened]... ple, something can be an EXACT irrational number in magnitude? -or is that just always nonsense?

Well yes, but then re-labelling 1/2 to be 1 we have just created an integer out of a fraction.

Come to think of it, it is possible to re-label all the rational numbers as integers (as they have the same 'size'ðŸ˜‰, but not so with the real numbers. Is this perhaps significant?