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I'm extremely old fashioned.

Prometheus had stated that this theorem is only true for an idealized triangle and not a physical triangle and what he meant by this is, at the atomic level, the dimensions of an actual object are always fluctuating.

But I would like to point out that does not mean that the theorem is not true. The theorem still holds because it is not concerned with these fluctuations in space time. Lack of precise measurements does not translate to lack of accuracy of the theorem. The theorem will give an accurate output per any input. The output may simply not represent the dimension of the respective side of the triangle at a later point in time. Additionally, while typically not practical to measure, the Heisenberg uncertainty principal states that we can know the dimensions of a physical triangle precisely, we just can't simultaneously know it's velocity as a frustrated student hurls it through the air.

Of course the the theorem is true, at least in a Euclidean metric. What I meant was that no physical triangle can ever BE a real triangle because of the fluctuations you mention. The theorem is true, but only of an idealised Platonic form of a triangle. Of course physical triangles can be a very good approximation to this ideal, but they can never reach it.

But a physical triangle can be a "real" triangle. Whoever told you it couldn't was likely using an analogy to drive home the point that, at the atomic and sub atomic level, matter is very dynamic, much like algebra instructors often initially tell students that certain second degree polynomials don't factor when they can all be factored, but the factoring method is beyond the scope of the course.

In accordance with the Heisenberg uncertainty principal, we can know the position of the corners of the triangle with certainty at any particular moment. Once we complete the calculation, we get the proper output for the unknown at the time of measurement which can also serve as an approximation at any other time but may still be the actual value.

We are assuming only small fluctuations in environmental conditions of course.

It doesn't have anything to do with Heisenberg's Uncertainty Principle. It's simply obvious that as one 'zooms in' to any 'straight' line, there will eventually appear deviations from this 'straightness', especially insofar as at the atomic level the line ceases to be a continuous distribution of points anyway. If there are no such entities as infinitesimal points in the physical, empirical world, then there can be no such thing as true lines or line segments, in which case there can be no triangles.

kokopelli

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Prometheus had stated that this theorem is only true for an idealized triangle and not a physical triangle and what he meant by this is, at the atomic level, the dimensions of an actual object are always fluctuating.

But I would like to point out that does not mean that the theorem is not true. The theorem still holds because it is not concerned with these fluctuations in space time. Lack of precise measurements does not translate to lack of accuracy of the theorem. The theorem will give an accurate output per any input. The output may simply not represent the dimension of the respective side of the triangle at a later point in time. Additionally, while typically not practical to measure, the Heisenberg uncertainty principal states that we can know the dimensions of a physical triangle precisely, we just can't simultaneously know it's velocity as a frustrated student hurls it through the air.

Of course the the theorem is true, at least in a Euclidean metric. What I meant was that no physical triangle can ever BE a real triangle because of the fluctuations you mention. The theorem is true, but only of an idealised Platonic form of a triangle. Of course physical triangles can be a very good approximation to this ideal, but they can never reach it.

But a physical triangle can be a "real" triangle. Whoever told you it couldn't was likely using an analogy to drive home the point that, at the atomic and sub atomic level, matter is very dynamic, much like algebra instructors often initially tell students that certain second degree polynomials don't factor when they can all be factored, but the factoring method is beyond the scope of the course.

In accordance with the Heisenberg uncertainty principal, we can know the position of the corners of the triangle with certainty at any particular moment. Once we complete the calculation, we get the proper output for the unknown at the time of measurement which can also serve as an approximation at any other time but may still be the actual value.

We are assuming only small fluctuations in environmental conditions of course.

It doesn't have anything to do with Heisenberg's Uncertainty Principle. It's simply obvious that as one 'zooms in' to any 'straight' line, there will eventually appear deviations from this 'straightness', especially insofar as at the atomic level the line ceases to be a continuous distribution of points anyway. If there are no such entities as infinitesimal points in the physical, empirical world, then there can be no such thing as true lines or line segments, in which case there can be no triangles.

In mathematics, we deal with mathematical spaces. Heisenberg's Uncertainty Principal does not apply there.

In the real world, we approximate. There are always upper and lower bounds to what we can actually measure.

For example, in mathematics pi is an irrational number. In the real world, anything more than a couple hundred decimal spaces are so (I don't know the precise number) is meaningless.

Prometheus had stated that this theorem is only true for an idealized triangle and not a physical triangle and what he meant by this is, at the atomic level, the dimensions of an actual object are always fluctuating.

But I would like to point out that does not mean that the theorem is not true. The theorem still holds because it is not concerned with these fluctuations in space time. Lack of precise measurements does not translate to lack of accuracy of the theorem. The theorem will give an accurate output per any input. The output may simply not represent the dimension of the respective side of the triangle at a later point in time. Additionally, while typically not practical to measure, the Heisenberg uncertainty principal states that we can know the dimensions of a physical triangle precisely, we just can't simultaneously know it's velocity as a frustrated student hurls it through the air.

Of course the the theorem is true, at least in a Euclidean metric. What I meant was that no physical triangle can ever BE a real triangle because of the fluctuations you mention. The theorem is true, but only of an idealised Platonic form of a triangle. Of course physical triangles can be a very good approximation to this ideal, but they can never reach it.

But a physical triangle can be a "real" triangle. Whoever told you it couldn't was likely using an analogy to drive home the point that, at the atomic and sub atomic level, matter is very dynamic, much like algebra instructors often initially tell students that certain second degree polynomials don't factor when they can all be factored, but the factoring method is beyond the scope of the course.

In accordance with the Heisenberg uncertainty principal, we can know the position of the corners of the triangle with certainty at any particular moment. Once we complete the calculation, we get the proper output for the unknown at the time of measurement which can also serve as an approximation at any other time but may still be the actual value.

We are assuming only small fluctuations in environmental conditions of course.

It doesn't have anything to do with Heisenberg's Uncertainty Principle. It's simply obvious that as one 'zooms in' to any 'straight' line, there will eventually appear deviations from this 'straightness', especially insofar as at the atomic level the line ceases to be a continuous distribution of points anyway. If there are no such entities as infinitesimal points in the physical, empirical world, then there can be no such thing as true lines or line segments, in which case there can be no triangles.

In mathematics, we deal with mathematical spaces. Heisenberg's Uncertainty Principal does not apply there.

In the real world, we approximate. There are always upper and lower bounds to what we can actually measure.

That's exactly what I said.

Prometheus had stated that this theorem is only true for an idealized triangle and not a physical triangle and what he meant by this is, at the atomic level, the dimensions of an actual object are always fluctuating.

But I would like to point out that does not mean that the theorem is not true. The theorem still holds because it is not concerned with these fluctuations in space time. Lack of precise measurements does not translate to lack of accuracy of the theorem. The theorem will give an accurate output per any input. The output may simply not represent the dimension of the respective side of the triangle at a later point in time. Additionally, while typically not practical to measure, the Heisenberg uncertainty principal states that we can know the dimensions of a physical triangle precisely, we just can't simultaneously know it's velocity as a frustrated student hurls it through the air.

Of course the the theorem is true, at least in a Euclidean metric. What I meant was that no physical triangle can ever BE a real triangle because of the fluctuations you mention. The theorem is true, but only of an idealised Platonic form of a triangle. Of course physical triangles can be a very good approximation to this ideal, but they can never reach it.

But a physical triangle can be a "real" triangle. Whoever told you it couldn't was likely using an analogy to drive home the point that, at the atomic and sub atomic level, matter is very dynamic, much like algebra instructors often initially tell students that certain second degree polynomials don't factor when they can all be factored, but the factoring method is beyond the scope of the course.

In accordance with the Heisenberg uncertainty principal, we can know the position of the corners of the triangle with certainty at any particular moment. Once we complete the calculation, we get the proper output for the unknown at the time of measurement which can also serve as an approximation at any other time but may still be the actual value.

We are assuming only small fluctuations in environmental conditions of course.

It doesn't have anything to do with Heisenberg's Uncertainty Principle. It's simply obvious that as one 'zooms in' to any 'straight' line, there will eventually appear deviations from this 'straightness', especially insofar as at the atomic level the line ceases to be a continuous distribution of points anyway. If there are no such entities as infinitesimal points in the physical, empirical world, then there can be no such thing as true lines or line segments, in which case there can be no triangles.

In mathematics, we deal with mathematical spaces. Heisenberg's Uncertainty Principal does not apply there.

In the real world, we approximate. There are always upper and lower bounds to what we can actually measure.

For example, in mathematics pi is an irrational number. In the real world, anything more than a couple hundred decimal spaces are so (I don't know the precise number) is meaningless.

We do approximate, as we only need "good enough". However Prometheus's perception that the Pythagorean Theorem does not apply to physical triangles as more than an approximation is incorrect. It applies because the corners do have a location that are

**allowed to be known**in accordance with the Heisenberg uncertainty principal.

It is not an issue of limits, as you think, though I understand why you think it is.

What we can't do, and I believe this is analogous to his perception, is claim that a physical triangle, no matter how thin, is a two dimensional object in three dimensional space. In that case, the two dimensional triangle is intangible and the physical triangle is tangible.

Anyway, if I am going to be brutally honest, the understanding of the concept in question is not something most come by casually and even Heisenberg first had some misunderstandings of his own principal, utilizing an observer effect, which has turned out to be false but has unfortunately been propagated in many undergraduate physics textbooks.

Last edited by Chronos on 05 Sep 2018, 3:58 pm, edited 2 times in total.

The issue has nothing, as I've already said, to do with Heisenberg's Uncertainty Principle or any other physical principle. The point I'm making is a metaphysical point, namely, that of the difference between the appearance of a thing and the thing itself. Physical points can be invoked in aid of this metaphysical point, as I've done above, but it isn't fundamentally a question of empirical science.

I never said that Pythagoras is not true of any physical triangle, but rather that no physical 'triangle' is a triangle. The mistake you're making is a category mistake of failing to observe the distinction between a conceptual construction and its physical representation.

I never said that Pythagoras is not true of any physical triangle, but rather that no physical 'triangle' is a triangle. The mistake you're making is a category mistake of failing to observe the distinction between a conceptual construction and its physical representation.

But you did say exactly what you claim to not have said. Here it is below.

But we all mis-speak sometimes, so if what you said does not accurately reflect your thoughts, then that is understood.

I am not a practitioner of the metaphysical, only the physical, and that took many years of study. You will have to define "metaphysical" to me as you understand it such that we are on the same page.

But you did say exactly what you claim to not have said. Here it is below.

I used the word 'physical' in the passage you quote here in the sense of 'empirical', not on the sense of 'relating to physics'. Two completely different meanings.

I am not a practitioner of the metaphysical, only the physical, and that took many years of study. You will have to define "metaphysical" to me as you understand it such that we are on the same page.

I am a student of both physics AND philosophy; I don't myself see how either can be taken in isolation from the other.

The sense in which my point is metaphysical rather than physical is that my reasoning above doesn't necessitate any reference to empirical propositions but is true in virtue of reason alone.

I think the bottom line is that physical triangles only have the property of triangularity by, as it were, analogy. Think over all the right triangles you saw in textbooks at high school; clearly, they weren't REALLY right triangles, or in fact even triangles at all, insofar as sufficiently accurate measurement would have shown them not to possess the properties predicated of a triangle.

I suggest that you do some research on Plato's theory of forms, as well as Kant's dichotomy of noumenon and phenomenon; both philosophers are hugely flawed, but there's no doubt pages on the net that can clarify these concepts with greater facility than I can.

But you did say exactly what you claim to not have said. Here it is below.

I used the word 'physical' in the passage you quote here in the sense of 'empirical', not on the sense of 'relating to physics'. Two completely different meanings.

As far as I am aware, physical in the sense as it relates to physics does imply empiricle. But if you intend to mean, or perceive it to be otherwise, I invite you to share your definitions.

I am not a practitioner of the metaphysical, only the physical, and that took many years of study. You will have to define "metaphysical" to me as you understand it such that we are on the same page.

I am a student of both physics AND philosophy; I don't myself see how either can be taken in isolation from the other.

Indeed many are. There is intersectionality, no doubt. But what is the purpose of thought but for pleasure if it never leads one to conclusions?

I think the bottom line is that physical triangles only have the property of triangularity by, as it were, analogy. Think over all the right triangles you saw in textbooks at high school; clearly, they weren't REALLY right triangles, or in fact even triangles at all, insofar as sufficiently accurate measurement would have shown them not to possess the properties predicated of a triangle.

I understand you are trying to express the idea that they are imperfect. I recognize this, as you are using this idea as a tool to communicate a point....you have predicated your point on the idea of imperfect triangles, understood. We can set that aside as you can be assured I understand that which you were trying to communicate using the example of triangles.

However on to the subject of the triangles in a textbook actually being imperfect, if the lines have a thickness T, which they do, and contained within those lines is a line with a thickness, t, cannot t be chosen such that a perfect triangle exists?

Thank you for the suggestion, though I am sure I was familiar with them at one point. But I will refresh my memory in case they offer something to me now, which they previously did not and has thus, left me.

I am pretty old-fashioned myself and eccentric. After many years of trying to fit in and not enjoying it, I decided to be myself and myself is pretty eccentric. It has been less than a year since I discovered that autism is the root of much of my eccentricity. I am comfortable with that and think that eventually you will be too.

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*Eyes that watch the morning star*

usually shine brighter,

Arms held out to dark they say,

usually hold tighter.

usually shine brighter,

Arms held out to dark they say,

usually hold tighter.

Threnody, Dorothy Parker

as modified by David Tamulovich

Nevertheless, 'physical' in the sense of 'empirical' does not imply 'physical' in the sense of physics.

Not exactly, because a true triangle would have infinitely thin lines; lines in Euclidean geometry are one-dimensional. But I think I understand where you're going wrong now: yes, we can choose the dimensions of a triangle so as to satisfy Pythagoras, for instance, we choose sides with lengths 3, 4 and 5 or any of the infinite (I think) number of Pythagorean triples. All I'm saying is that while we can in a sense define such a triangle, and even think about it conceptually, we can never DEPICT it, or represent it in the external world in any way, except by imperfect analogy.

Indeed many are. There is intersectionality, no doubt. But what is the purpose of thought but for pleasure if it never leads one to conclusions?

Most philosophy is hot air, it's true. Its value lies, as Wittgenstein pointed out, in its clarifying our concepts; in its ability to show us which parts of our ideas are a vestige of language, and which are relevant to

*das Ding an sich*.

Thank you.

Nevertheless, 'physical' in the sense of 'empirical' does not imply 'physical' in the sense of physics.

Please state the definition of empirical you are using.

Not exactly, because a true triangle would have infinitely thin lines; lines in Euclidean geometry are one-dimensional.

If we take an interval of thickness x, which is finite, we can claim that within it exists an infinitesimally small thickness dx. That an infinitesimally small thickness does not actually exist in a discrete universe is irrelevent because the interval x does and contains all of the information needed to construct outputs as if it did. A similar phenomena occurs in waveform sampling during the process of digitization. We can take an anolog (time continuous) sound wave and digitize it by sampling at rate no less than the Nyquist rate. From these sample points we can construct a discrete version of the wave, and also reconstruct the analog wave, without fidelity loss. In other words, all of the information contained in the original waveform is preserved even though we did not sample all of the points in the analog wave. The implication of this is that some points don't matter; and if they don't actually matter, why insist they do? Why claim a theoretical right triangle that only exists mathematically is more valid of a right triangle than a physical triangle, within which it's boundaries exists a right triangle which also satisfies the conditions of a theoretical triangle?

Are you an adherent to the notion of absolute truth?

How would we be having this conversation if I were wrong? Are the principals on which devices are designed and built wrong if the devices work?

Does it not exist when it is contained within a physical triangle? Does it only exist if alone?

Indeed many are. There is intersectionality, no doubt. But what is the purpose of thought but for pleasure if it never leads one to conclusions?

Most philosophy is hot air, it's true. Its value lies, as Wittgenstein pointed out, in its clarifying our concepts; in its ability to show us which parts of our ideas are a vestige of language, and which are relevant to

*das Ding an sich*.

With the exception of the field of logic, and I mean that in the same sence that an academic catalogue would when speaking of a logic course, where one would learn how to reduce situations to symbols and follow a logical process to a conclusion, and certain thought experiments predicated on that logic, intuitively or otherwise, I have not seen the field of philosophy ever clarify much; rather, more of the opposite seems to come from it.

Mathematics is important no doubt. It can teach us much about the universe that we cannot always readily observe. But it has only done so when we can observe that the predictions hold in the physical world.

If you are interested in logic, and have not read it already, I highly recommend a fun little book called "The Lady or the Tiger & Other Logic Puzzles", by Raymond M. Smullyan

I also recommend you familiarize yourself with the author and his other works if you are not already aware of him and his publications, as you might find something relavent to yourself in them.

Unfortunately he passed away last year. You can purchase the book but I believe the book can be found for download illegally in pdf form. I leave it to you to determine whether or not it is ethical to download the pdf for free.

On one hand, you would be duplicating the information such that you are constructing your own "object", not taking an "object" from someone else in such a way that deprives them of it. On the other hand, you are stealing intellectual property. Yet on the other hand, dead men need no resources.

Most of the money you spend will go to where you purchase the book from, and if you purchase an old or used version, none of the money will go to the publisher. It merely goes to the seller and shipper, and may disperse throughout the economy.

If you purchase an electronic version, the seller recieves all of the money (and may disperse it) except perhaps a percentage set aside for the estate of the author. The seller could probably provide the book for free. Yet on the other hand, royalties that end up in the author's estate may fund grants, scholarships and endowments that enable others to contribute valuable things to society, though maybe not in a way that benefits you. But then again they may also go to his family members and they may just by cigarettes with them.

You can also avoid this ethical problem by checking the book out from the library, even in e form from an online one, but would have to wonder, from an ethical stand point, how that differs from illegally downloading the pdf for free.

I will leave all of that for you to answer.

By "empirical", in the original passage, I meant "existing as an object in the physical world".

I entirely agree; and of course, this is why physics is "useful" in the vulgar sense and mathematics is only "useful" in virtue of its applications to physics or other scientific fields. It doesn't change the fact that there is nothing we can point to in the physical world which has the property of "triangularity" as part of its nature. From what you're saying, and I think you're right, we "willingly suspend disbelief" in the "atriangularity" of the object depicted on the sheet of paper in order to derive "useful" conclusions about it. Of course, this is true, and it has its virtues, but that doesn't change the fact that the object depicted on the sheet of paper is, as it were, a shadow of the subject in question - that the triangle itself is a conceptual or mental entity.

For the sake of intellectual and linguistic rigour. You mentioned the field of logic, and this is the same reason why in logic we distinguish between words and the things they represent. For instance, a logician will tell you that the following sentence is valid:

**Quote:**

While the next sentence is invalid:

**Quote:**

A word, such as "London", may or may not have any given number of letters; a city cannot be said to have any number of letters as letters and cities are conceptually incompatible. Of course the practise is irrelevant for most purposes and the average man, in going about his business, will readily interchange the two without a second thought - and that's fine, because everyone knows what he means when he chooses the stricly incorrect alternative. In science, however, greater rigour is called for.

Yes, though I don't claim to know what it is on most subjects.

RE Richard Smullyan, I know very roughly who he is and have been meaning since seeing his obituary to read some of his books, but haven't had time so far.

As far as piracy is concerned, I reject it as unethical, though as far as unethical acts go, it seems to me a moderately trivial one.

By "empirical", in the original passage, I meant "existing as an object in the physical world".

I entirely agree; and of course, this is why physics is "useful" in the vulgar sense and mathematics is only "useful" in virtue of its applications to physics or other scientific fields. It doesn't change the fact that there is nothing we can point to in the physical world which has the property of "triangularity" as part of its nature. From what you're saying, and I think you're right, we "willingly suspend disbelief" in the "atriangularity" of the object depicted on the sheet of paper in order to derive "useful" conclusions about it. Of course, this is true, and it has its virtues, but that doesn't change the fact that the object depicted on the sheet of paper is, as it were, a shadow of the subject in question - that the triangle itself is a conceptual or mental entity.

For the sake of intellectual and linguistic rigour. You mentioned the field of logic, and this is the same reason why in logic we distinguish between words and the things they represent. For instance, a logician will tell you that the following sentence is valid:

**Quote:**

While the next sentence is invalid:

**Quote:**

A word, such as "London", may or may not have any given number of letters; a city cannot be said to have any number of letters as letters and cities are conceptually incompatible. Of course the practise is irrelevant for most purposes and the average man, in going about his business, will readily interchange the two without a second thought - and that's fine, because everyone knows what he means when he chooses the stricly incorrect alternative. In science, however, greater rigour is called for.

Yes, though I don't claim to know what it is on most subjects.

RE Richard Smullyan, I know very roughly who he is and have been meaning since seeing his obituary to read some of his books, but haven't had time so far.

As far as piracy is concerned, I reject it as unethical, though as far as unethical acts go, it seems to me a moderately trivial one.

In that case, you can borrow the book in electronic form (I am assuming legally) from Open Library

You might find some of the first few puzzles trivial but there are varying levels of complexity, with puzzles most would find difficult in the later pages of the book.

Mona Pereth

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To Prometheus18: I am curious about your username. If you feel inclined to tell us about it, what does the ancient Greek Titan Prometheus mean to you personally?

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