"You Can’t Prove A Negative"

[bytesplicer]

Indeed, and proving something to be useful is an easier job than proving that it is right. This reflects for me why science is slowly replacing religion, not because it says anything about an absolute truth, but because it is so much more useful.

[Bill Wiltrack]

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No big deal…

Not even sure who first stated that there is a square root of 2 but that statement is false.

The square root of two is what is called an irrational number.


There is no known square root of 2.


Today the common truncated value is taken to 65 decimal places.


Just because the square root of 2 is irrational does not mean we throw out all that is mathematical, far from it.

Math is an indispensable science to the evolution of mankind.


Just because someone misstated a fact does not mean that all of what that member posts is deceitful.

Far from it, we all make mistakes.


And perhaps, in the context of this thread, all of our individual misconceptions, beliefs, and misstatements play a part in the approach or conclusions of the thread You Can’t Prove a Negative.



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[chaz wyman]

Indeed, and proving something to be useful is an easier job than proving that it is right. This reflects for me why science is slowly replacing religion, not because it says anything about an absolute truth, but because it is so much more useful.

Well said. And we ought to beware of people who seek to use science as a kind of religion by pretending that it has ultimate answers to questions that are ultimately meaningless.

[evangelicalhumanist]

There is no known square root of 2.

I would not say quite that. I would agree that there is no exhaustively expressible square root of 2, either as a sequence of decimal digits nor as a fraction. However, if we accidently build a right triangle out of platinum wire half an angstrom thick, and the sides around the right angle are both 1 unit (any sort of unit, I don't care) in length, then the hypoteneuse, whether we can accurately describe it or not, is root*2 units in length.

You may wish to argue that that is idealist, that in the real world we cannot manufacture such perfection, and again you would be correct. But I am satisfied that such objections are niggling and unimportant, that we can at least understand how a thing could be a physical expression of the square root of 2.

[Bill Wiltrack]

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There is no known square root of 2.


There is no exhaustively expressible square root of 2, either as a sequence of decimal digits nor as a fraction.

There is no known square root of 2.


Hypoteneuse is commonly spelled hypotenuse.


In the real world we cannot manufacture such imperfection and be correct.

Some may be satisfied that such objections are niggling and unimportant.

There is no known square root of 2.




Unless you are an evangelical spreading...whatever they spread around.


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[bytesplicer]

I'm with evangelical on root 2. Saying there is no root of 2 is not accurate (in the context of maths), we simply don't have a means of expressing it in numerical notation. Symbolic representation (or rather, further abstraction of the symbols we call numbers) on the other hand, handles root 2 without problem in the form of surds or fractional exponents, you can't write out the number, but you can use it in calculations. Similarly with pi and i. There's also the question of accuracy versus equality. If 40 digits of pi are enough to measure the universe to within an atoms width, for all intents and purposes you can say the 40 digit version is pi, though strictly speaking it isn't. As chaz says earlier, this is not truth, but it is useful.

[chaz wyman]

There is no known square root of 2.

I would not say quite that. I would agree that there is no exhaustively expressible square root of 2, either as a sequence of decimal digits nor as a fraction. However, if we accidently build a right triangle out of platinum wire half an angstrom thick, and the sides around the right angle are both 1 unit (any sort of unit, I don't care) in length, then the hypoteneuse, whether we can accurately describe it or not, is root*2 units in length.

Actually this proves nothing of the kind. The hypotenuse is NOT the Sqrt of 2, because this entire exercise demonstrates that there are no integers in nature and thus the Pythagorean relationship between sides is an approximate relationship, limited by the facts that Maths models reality but does not represent it. maths is useful, not natural.
Nature is the essence, maths follows it. We are not discovering maths which is out there existing in the universe but imposing a model of our own invention to help us understand it.

You may wish to argue that that is idealist, that in the real world we cannot manufacture such perfection, and again you would be correct. But I am satisfied that such objections are niggling and unimportant, that we can at least understand how a thing could be a physical expression of the square root of 2.

No you are a naive realist, not an idealist. An idealist realises that we construct our reality from our own models, a realist thinks that those models are what reality is.

[chaz wyman]

There is no known square root of 2.

I would not say quite that. I would agree that there is no exhaustively expressible square root of 2, either as a sequence of decimal digits nor as a fraction. However, if we accidently build a right triangle out of platinum wire half an angstrom thick, and the sides around the right angle are both 1 unit (any sort of unit, I don't care) in length, then the hypoteneuse, whether we can accurately describe it or not, is root*2 units in length.

Actually this proves nothing of the kind. The hypotenuse is NOT the Sqrt of 2, because this entire exercise demonstrates that there are no integers in nature and thus the Pythagorean relationship between sides is an approximate relationship, limited by the facts that Maths models reality but does not represent it. maths is useful, not natural.
Nature is the essence, maths follows it. We are not discovering maths which is out there existing in the universe but imposing a model of our own invention to help us understand it.

You may wish to argue that that is idealist, that in the real world we cannot manufacture such perfection, and again you would be correct. But I am satisfied that such objections are niggling and unimportant, that we can at least understand how a thing could be a physical expression of the square root of 2.

No you are a naive realist, not an idealist. An idealist realises that we construct our reality from our own models, a realist thinks that those models are what reality is.

[evangelicalhumanist]

No you are a naive realist, not an idealist. An idealist realises that we construct our reality from our own models, a realist thinks that those models are what reality is.

Okay, I give up. I don't know what we're arguing about, but using words like "naive" in a polite conversation changes the tone slightly.

I am perfectly well aware that all of our reality -- ALL OF IT -- is constructed from our own models. But what do you gain by carrying that to the extreme of denying our modeled reality altogether? When you deny all the tools that we use to describe the universe in a way such that we can use it and live in it, what do you have left to help you understand the world you actually do live in?

Isn't that, after all, what Hume meant when he said that though we have rational grounds for believing in an objective reality, we also have no choice but to act as if is true? What will we gain by acting as if it were not?

And therefore, we have a number which we can describe perfectly well (that which, when multiplied by itself yields a result of 2), which describes the relationship of between the sides of a very simple (the simplest) right triangle. We have another number (that which, when multiplied by itself yields a result of 4). Does that bother you as much?

I guess what I'm getting at is --- what are you getting at?

[bytesplicer]

Actually this proves nothing of the kind. The hypotenuse is NOT the Sqrt of 2, because this entire exercise demonstrates that there are no integers in nature and thus the Pythagorean relationship between sides is an approximate relationship, limited by the facts that Maths models reality but does not represent it. maths is useful, not natural.
Nature is the essence, maths follows it. We are not discovering maths which is out there existing in the universe but imposing a model of our own invention to help us understand it.

You are right when you say a 'perfect' right angle triangle cannot be constructed or occur naturally, but I disagree on some of your reasoning. Mathematics occupies the cusp between realism and idealism. The pythagorean relationship between sides isn't an approximation per se (though it acts like one), it represents the maximum accuracy with which such a relationship can be measured. As with my statement earlier about 40 digits of pi, we can build right-angle triangles, we do it all the time to high degrees of accuracy that approach but will never reach the maximum accuracy described by the relationship. We approach this hard limit through increasing the resolution of our tools, and, as resolution increases, mathematics describes what we see in nature ever more accurately. To say it is just a human conceit, as opposed to a mapping of patterns that occur in nature, is to pre-suppose we could construct our maths a different way and still achieve the accuracy we do. This is not clear enough at the moment to make that kind of judgement. On the other hand, saying there is maths out there 'existing' is also not right, but the patterns that we see in various phenomenon do inevitably lead to the mathematical relations we use.

Maths, patterns in nature that we uncover, or models we use to describe what we see? Either way, at increasing resolutions maths can describe nature almost perfectly. Useful, and increasingly useful as resolution increases, but never perfect. Perhaps that is how nature operates, and maths constitutes the fullest description, perhaps not. We don't know either way yet.

[chaz wyman]

Actually this proves nothing of the kind. The hypotenuse is NOT the Sqrt of 2, because this entire exercise demonstrates that there are no integers in nature and thus the Pythagorean relationship between sides is an approximate relationship, limited by the facts that Maths models reality but does not represent it. maths is useful, not natural.
Nature is the essence, maths follows it. We are not discovering maths which is out there existing in the universe but imposing a model of our own invention to help us understand it.

You are right when you say a 'perfect' right angle triangle cannot be constructed or occur naturally, but I disagree on some of your reasoning. Mathematics occupies the cusp between realism and idealism. The pythagorean relationship between sides isn't an approximation per se (though it acts like one), it represents the maximum accuracy with which such a relationship can be measured.

I kinda knew you would say that. That is a claim that can never be verified. It is one of the unspoken axioms upon which maths stands. What it actually says is that if only nature were more like maths then this would be real. The point being that maths is a self justifying system which does not rely on nature or evidence at all. There is no right angle in nature, perfect of not.

As with my statement earlier about 40 digits of pi, we can build right-angle triangles, we do it all the time to high degrees of accuracy that approach but will never reach the maximum accuracy described by the relationship. We approach this hard limit through increasing the resolution of our tools, and, as resolution increases, mathematics describes what we see in nature ever more accurately. To say it is just a human conceit, as opposed to a mapping of patterns that occur in nature, is to pre-suppose we could construct our maths a different way and still achieve the accuracy we do. This is not clear enough at the moment to make that kind of judgement. On the other hand, saying there is maths out there 'existing' is also not right, but the patterns that we see in various phenomenon do inevitably lead to the mathematical relations we use.

Maths, patterns in nature that we uncover, or models we use to describe what we see? Either way, at increasing resolutions maths can describe nature almost perfectly. Useful, and increasingly useful as resolution increases, but never perfect. Perhaps that is how nature operates, and maths constitutes the fullest description, perhaps not. We don't know either way yet.

[Impenitent]

I kinda knew you would say that. That is a claim that can never be verified. It is one of the unspoken axioms upon which maths stands. What it actually says is that if only nature were more like maths then this would be real. The point being that maths is a self justifying system which does not rely on nature or evidence at all. There is no right angle in nature, perfect of not.

circular reasoning (self justifying) as "proven" truth is far more common

a closed system of referents that refer to no empirically observable events... mathematics

a closed system of referents that refer to empirically observable events... language

truth based on a closed system of referents is true definitionally

-Imp

[bytesplicer]

I kinda knew you would say that. That is a claim that can never be verified. It is one of the unspoken axioms upon which maths stands.


I don't think we disagree here, I get the impression you've made up your mind more in favour of idealism whereas I'm still on the fence, waiting to see how close maths can come to describing nature.

What it actually says is that if only nature were more like maths then this would be real.

Looking from the other side, if maths progresses to be more like nature (i.e. describes it more fully), shows this isn't cut and dry, at least not yet. Mathematical discoveries (or new models, as you prefer) are ongoing.

The point being that maths is a self justifying system which does not rely on nature or evidence at all.


This is true, but maths does look to nature for verification, at least the maths that looks to be accepted as a description of nature.

There is no right angle in nature, perfect of not.

This of course depends on whether you think man-made is natural or not. Also, electromagnetic radiation involves fields at right angles to each other, though again whether this is a product of idealism and how we visualise it, or a real property, is debatable. There's also the issue of scale, we haven't zoomed in or out enough to know for sure. Our intuition that it isn't is of course based on mathematics, so whether you want to rely on that to make the final call depends on your trust in maths. Only observation will tell, and we may never observe such a thing, bringing us squarely back on topic about proving a negative...

[chaz wyman]

I kinda knew you would say that. That is a claim that can never be verified. It is one of the unspoken axioms upon which maths stands.


I don't think we disagree here, I get the impression you've made up your mind more in favour of idealism whereas I'm still on the fence, waiting to see how close maths can come to describing nature.

What it actually says is that if only nature were more like maths then this would be real.

Looking from the other side, if maths progresses to be more like nature (i.e. describes it more fully), shows this isn't cut and dry, at least not yet. Mathematical discoveries (or new models, as you prefer) are ongoing.

The point being that maths is a self justifying system which does not rely on nature or evidence at all.


This is true, but maths does look to nature for verification, at least the maths that looks to be accepted as a description of nature.

There is no right angle in nature, perfect of not.

This of course depends on whether you think man-made is natural or not.

Yes and no. The point I was making is that a mathematical rt angle is an ideal (and thus perfect), in nature we are able to create approximations. No physical rt angle is perfect.

Also, electromagnetic radiation involves fields at right angles to each other, though again whether this is a product of idealism and how we visualise it, or a real property, is debatable.

This is not debatable. I fail to see why EMR is a special case.

There's also the issue of scale, we haven't zoomed in or out enough to know for sure. Our intuition that it isn't is of course based on mathematics, so whether you want to rely on that to make the final call depends on your trust in maths. Only observation will tell, and we may never observe such a thing, bringing us squarely back on topic about proving a negative...

I don't think we are in disagreement, but I don't see what you mean by bringing us back to proving a negative.????

[bytesplicer]

I kinda knew you would say that. That is a claim that can never be verified. It is one of the unspoken axioms upon which maths stands.


I don't think we disagree here, I get the impression you've made up your mind more in favour of idealism whereas I'm still on the fence, waiting to see how close maths can come to describing nature.

What it actually says is that if only nature were more like maths then this would be real.

Looking from the other side, if maths progresses to be more like nature (i.e. describes it more fully), shows this isn't cut and dry, at least not yet. Mathematical discoveries (or new models, as you prefer) are ongoing.

The point being that maths is a self justifying system which does not rely on nature or evidence at all.


This is true, but maths does look to nature for verification, at least the maths that looks to be accepted as a description of nature.

There is no right angle in nature, perfect of not.

This of course depends on whether you think man-made is natural or not.

Yes and no. The point I was making is that a mathematical rt angle is an ideal (and thus perfect), in nature we are able to create approximations. No physical rt angle is perfect.

Yeah, that's cool, we do agree.

Also, electromagnetic radiation involves fields at right angles to each other, though again whether this is a product of idealism and how we visualise it, or a real property, is debatable.

This is not debatable. I fail to see why EMR is a special case.

This was thinking out loud. An electromagnetic field 'appears', at least from what I know, to be an example of a perfect right angle in nature. My confusion arises because, as one of the themes of this discussion shows, we represent things in ways that don't always correspond exactly to reality, like the shell model of atoms, which encapsulates information but doesn't describe exactly what is happening. I'm not sure if EMR is the same in this respect, with the right angle of the fields not being a right angle in reality. I actually don't know. Also, could a right-angle in a EM field be classed as physical anyway.

There's also the issue of scale, we haven't zoomed in or out enough to know for sure. Our intuition that it isn't is of course based on mathematics, so whether you want to rely on that to make the final call depends on your trust in maths. Only observation will tell, and we may never observe such a thing, bringing us squarely back on topic about proving a negative...

I don't think we are in disagreement, but I don't see what you mean by bringing us back to proving a negative.????

This boils down to me agreeing there are no right angle (triangles) in nature, but not quite understanding why I agree. How is this known? Is the proof based on mathematics, or something else? It seems like a negative that has been proven, but I can't recall what the proof is or even if there is one.