fishfry

Have we really proved the existence of irrational numbers?
Metaphysicians, being trained in this field, are best able to say whether something exists or not. — Metaphysician Undercover

Ahhhhh, so we shouldn't poll the general public as @Ryan suggests; nor the mathematicians, which I suggest; but rather the metaphysicians! Well that certainly makes a big difference. /s

So in the end you agree with the notion that existence is contingent on opinion, and you simply differ on which opinions count. You just lost the argument methinks.

And what if I find a metaphysician who, based on two years of dialog with me, clearly hasn't bothered to learn the most elementary facts of mathematics? Why should I trust that individual's judgment about anything?
The paradox of Gabriel's horn.
An infinite acceleration is required to go from rest to moving. — Metaphysician Undercover

No it's not. When you get in your car and start driving to the store, do you experience infinite acceleration? What's that feel like, exactly? According to special relativity, you should be pressed against the back of your seat with infinite force. You'd be crushed before you drove a foot. What do you say?

ps -- Let's do the math. Say I'm at rest and start moving at 1 unit/second or whatever. In physics we need to give the units but in math we'll just say the velocity is 1. So at 1 second we've gone 1 unit, at 2 seconds we've gone two units, etc.

So our position function is p(t) = t; and our velocity is always 1, which is consistent with the first derivative of position being the derivative of t with respect to t, or 1.

Now this is tricky and this is where you got yourself confused. What was our instantaneous acceleration at 0? After all we weren't moving and then a tiny moment later we were. Well, the graph of our position look like this:
```
             /
           /
----------o
```
This function is not differentiable at zero. There is no instantaneous velocity at zero and no definite acceleration either. I agree that this is counterintuitive, and your intuition is not uncommon. But it's wrong. Clearly it's wrong. If you experienced infinite acceleration even for a moment, every atom in your body would be flattened like so many pancakes.

I actually agree with you about the intuition. If we're not moving, how do we start moving? It's a bit of a mystery actually, I'm not sure what physicists say about this. Well I guess I do know. If we're a steel ball in Newton's cradle, or we're a ball on a pool table, we start moving when we get smacked by another ball that transfers its momentum to us. But how does our velocity go instantaneously from zero to nonzero? The Newtonian physics works out, but not the intuition.

resolving the problem of how the non-dimensional truly relates to the dimensional. — Metaphysician Undercover

A little woo-woo-y there @MU. By the way, how do the zero-dimensional, zero-length points in the unit interval make up the one-dimensional, length 1 unit interval? That's actually another mystery, despite the fact that we have a mathematical formalism that says $\int_0^1 dx = 1$ . The math works but we have no metaphysical explanation that I know of.
Have we really proved the existence of irrational numbers?
Why give special status to " a preponderance of mathematicians", granting them the capacity to determine the existence of things? Isn't there a preponderance of Catholic theologians who believe in Jesus, and a preponderance of Pastafarians who believe in the existence of the spaghetti monster? Why do you think that mathematicians, simply by believing in something have the capacity to grant that something "existence", while for other groups of people, simply believing in something is insufficient for the existence of what is believed in? — Metaphysician Undercover

You ask a good question, and one I can't answer. This is the only thing you've ever said to me that has made me stop and think, and for which I have no good answer.

My standard response is that math is a formal game, like chess. A position is legal if and only if it follows from the rules, there's no right or wrong to it, nor any deeper reason.

But I must admit that math isn't really that simple. 5 is prime, and that seems to be true independently of the opinions of people. Mathematical truth has a necessity that's forced on us in some way nobody can understand.

Bowling balls fall down, and that's forced on us too, but bowling balls are physical. Mathematical objects are purely abstract entities, yet the facts about them seem absolutely true independent of their discovery.

I haven't got a good answer. A lot of smarter people than I don't have a good answer either. Have you?
How to have a fulfilled life accepting that it will end someday without knowing when and how?
I want to go like my grandpa did, peacefully in his sleep.. Not screaming in terror like his passengers.
Have we really proved the existence of irrational numbers?
I meant to write "[...] does not require [...]." I edited my post just now upon reading your post and realizing that I mistakenly left out the word 'not'. — GrandMinnow

Doh! That would have saved me a lot of typing! LOL.
Have we really proved the existence of irrational numbers?
Understanding the sense of mathematical existence statements - such as the existence of irrational numbers - does require subscribing to mathematical platonism. — GrandMinnow

This I would disagree with. One can take the viewpoint that symbolic math is a human endeavor; and that a thing has mathematical existence whenever a preponderance of mathematicians agree that it does. No Platonism needed. We've seen this standard applied to irrational numbers, negative numbers, transcendental numbers, complex numbers, quaternions, transfinite numbers, and many other now-familiar mathematical objects.

In order to demonstrate that sqrt(2) has mathematical existence, I do not need to posit a mystical Platonic realm in which sqrt(2) lives. If I did, I might be challenged: What else lives there? The baby Jesus? The Flying Spaghetti Monster? Pegasus the flying horse? No, I don't need to sort all this out just to know that sqrt(2) exists.

I can show sqrt(2) exists as others mentioned, as real number such that $x^2 = 2$ , once I've formalized the construction of the real numbers and shown their completeness.

We can simply make up an arbitrary symbol $\sqrt 2$ with the property that $(\sqrt 2)^2 = 2$ , and then consider the collection of all rational numbers $a + b \sqrt 2$ where $a$ and $b$ are rational. We will find that we've invented a set of numbers that obey all the rules of a field (I can add, subtract, multiply, and divide with all the usual properties) that contains a square root of 2. This works very nicely.

In fact this is exactly how we introduce the complex numbers to students, as the set of all $a + bi$ where $a$ and $b$ are real and $i^2 = 1$ .

We can formalize the above idea by starting from the rational numbers, forming the ring of polynomials with rational coefficients, and mod out the ideal generated by the polynomial $x^2 - 2$ . The resulting object is a field in which 2 has a square root.

In other words the two previous paragraphs show that if you believe in the rational numbers, you can easily adjoin to them a square root of 2. That is: if the rationals exist then so does the square root of 2. One does not need any Platonic realm to perform these symbolic constructions.

If someone asks if sqrt(2) as I've defined it has mathematical existence, I just point to any textbook on real analysis or abstract algebra. I do not need a book on metaphysics! All the mathematicians in the world agree that these constructions are valid and that's what gives sqrt(2) mathematical existence.

Now let me give an objection to what I said, one raised by @Metaphysician Undercover when we had this convo a while back. And that is, that mathematical existence is now contingent on what people say. My idea isn't any better than @Ryan's idea of polling all the humans to see if they're thinking of sqrt(2) at this very instant. My standard is to poll all the mathematicians in the world to see if most of them are willing to agree that sqrt(2) exists. I'd be the first to agree that my criterion for mathematical existence has some problems.

But a Platonic world where non-physical things exist? That seems untenable. So we're left with the opinions of mathematicians.
Have we really proved the existence of irrational numbers?
Are you proposing this as proof of the existence of God? — T Clark

Most definitely. With extra cheese.
Have we really proved the existence of irrational numbers?
A = sandwich * r^2. Doesn't work for me. Arccos (-1) = sandwich. Nope. — T Clark

The volume of a pizza of radius z and height a is pi z z a.
Do We Need Therapy? Psychology and the Problem of Human Suffering: What Works and What Doesn't?
unprompted by external political influences. — Maw

LOL. No. Nor does some corporate press release constitute evidence.
Have we really proved the existence of irrational numbers?
He only proved that √2 is not a rational number. He did not prove that √2 is an irrational number. Yes, I'm concerned with proof by contradiction. — Ryan O'Connor

Constructivists deny the law of the excluded middle. You might be interested in this. For my own part I don't have any affinity for constructivism although it's enjoying a resurgence lately due to the influence of computer science and computerized mathematical proof systems. Brouwer's revenge, I like to call it.

https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_mathematics)

If I flip a coin and tell you that it is not heads that is not proof that it is tails because it may have landed on its side. — Ryan O'Connor

Different issue. Landing on side should be included in the outcome space. In high school statistics we were flipping coins once and a nickel started rolling on its side and slowing down and darn near landed on its edge, but it hit the wall and fell over. But I almost saw it happen.

Actually, I want to strengthen it even more so to say that a number only exists when it is being computed. If there is no computer currently thinking about 3 right now then the number 3 does not exist as an actualized number. It only has the potential to exist. — Ryan O'Connor

I see your point, but then existence becomes contingent on what everyone's thinking about and/or computing. 3 might or might not exist depending on what 7 billion people and a few hundred million computers are thinking or computing at this exact instant. What kind of standard for existence is that? Not one that would have much support from ontologist, I'd wager.

I think this sort of view is required if we are to avoid actual infinity. — Ryan O'Connor

If you deny the mathematical existence of the natural numbers you not only deny ZF, but also Peano arithmetic. That doesn't let you get any nontrivial math off the ground. Not only no theory of the real numbers, but not even elementary number theory. You may be making a philosophical point but not one with much merit, since it doesn't account for how mathematics is used or for actual mathematical practice.

Otherwise, how would a constructivist answer the question: how many numbers are there? — Ryan O'Connor

Well then you're an ultrafinitist. You not only deny the existence of infinite sets; you deny the existence of sufficiently large finite sets.

I do actually have some sympathy for the ultrafinitist position, since it's the only mathematical ontology that is consistent with what's known about the physical world. But your particular flavor of ultrafinitism is untenable, granting existence to only those things that someone is thinking about or computing at any given instant. Under such a philosophy we can never say whether a given number or mathematical object exists.

My response to such a question is 'how many numbers are where? In what computer?' — Ryan O'Connor

Even the constructivists, with whom I've had many an interesting discussion in these very pages, believe in computable numbers. There is a countable infinity of them. Computable numbers, I mean, not constructivists.

The ultrafinitists don't put any particular upper limit on how large a number can be, only that there aren't infinitely many of them.

But you want to not only say that, but that whether a given number exists or not depends on whether someone's thinking of it. How can we ever determine that? It's an unverifiable standard. There is then no way to know whether any number exists and whether it still exists five minutes from now. It's impossible to hold such a view along with any kind of coherent ontology of numbers.

I believe that the decimal representation for 1/3 cannot exist but nevertheless the number certainly can. For example, it is LL on the Stern-Brocot tree. And we can do exact arithmetic using any rational number using the Stern-Brocot tree. — Ryan O'Connor

Why doesn't pi exist? It has a representation as a finite-length algorithm. By exist I mean mathematical existence of course, that's the only kind of existence I'm talking about.

But won't there always be undecidable statements? It seems like your definition is too restrictive as it would be missing some truths. — Ryan O'Connor

Well in any sufficiently interesting mathematical system we are always missing some truths. That's just a fact. But at least it's not contingent. The Continuum hypothesis is always undecidable in ZFC. Now and five minutes from now and five million years from now. And in ZFC + CH, it's provable. Now and five minutes from now. We have logical certainty about what exists, unlike with your system in which we have to constantly poll 7 billion people and several hundred million computers.

Why is it necessary to have a number system which is complete? — Ryan O'Connor

Because otherwise the real number line has holes in it. The intermediate value theorem is false. There's a continuous function that's less than zero at one point and greater than zero at another, but that is never zero. Of course the constructivists patch this up by limiting their attention to only computable functions. The constructivists have answers for everything, which I never find satisfactory.

I have no doubt that pi is irrational (i.e. not a rational number). But a sandwich is also not a rational number. My argument is that just as a sandwich is not a number, perhaps neither is pi. Maybe it's an algorithm. — Ryan O'Connor

Of course LEM is always relative to a given universe of discourse. If pi is a real number that's not rational, then it's a real number that's irrational. Without the restriction to real numbers, you're right. It could be a sandwich. Time for dinner.
Have we really proved the existence of irrational numbers?
But perhaps it is simplest if I take existence out of my original question: Have we really proved that √2 is an irrational number? — Ryan O'Connor

Yes, Pythagoras gets credit for (but probably didn't personally have anything to do with) the proof that sqrt(2) is irrational. Is it proof by contradiction that you're concerned about?

It's interesting that you agree that sqrt(2) is computable, as are pi, e, and every other mathematical constant that anyone can name. Except for Chaitin's constant, which we can name but which isn't computable.

You seem to be applying a much stricter standard than even the mathematical constructivists. They would allow the existence of any computable number, since we can give an algorithm to approximate it to any desired degree.

But you want to strengthen that standard to say that a number only exists when we can not only give an algorithm for it, but that we can execute the algorithm to completion given the physical constraints of the universe. That's a terribly restrictive standard, for example 1/3 = .3333... doesn't exist according to you.

But why not? Why are you privileging decimal notation? The algorithm for pi is finite and expresses pi exactly. So what if the decimal representation's not finite? Why should that be the standard? If we have an algorithm, we have the number. That's the constructivist point of view.

For my own part, I take mathematical existence to mean anything that can be proven logically using the axioms of any given mathematical system. That includes all the computable numbers and all the noncomputable ones as well, which after all are necessary to ensure the completeness of the real numbers. And no mathematical objects exist at all in the physical world, since they're all abstractions.
Have we really proved the existence of irrational numbers?
What do you mean by existence? Have we proved the existence of the number 3? Of 1/3 = .333...? Of -6? Of $i$ such that $i^2 = -1$ ? Of the quaternion $1 + i + j + k$ ? Of $\aleph_3$ ? Of $\omega_1$ , the first uncountable ordinal?

What does it mean (to you) to prove that a number exists?
The paradox of Gabriel's horn.
2) The limit of the journey corresponds to the final destination, which if anything would be (∞,0). — Ryan O'Connor

I surely disagree. There is no "final destination." That's @MU's error, why are you amplifying it?
Do We Need Therapy? Psychology and the Problem of Human Suffering: What Works and What Doesn't?
We could ask to what extent is despair a mental health problem? — Jack Cummins

Of course despair is a mental health problem. But a person experiencing despair need not be a nihilist nor vice versa. There are two distinct things: (1) How you feel; and (2) Crap you read. It's important not to confuse the two. Especially these days when the mainstream media is devoted to publishing crap that makes people feel bad, in order to drive clicks.

By the way the left wing Gestapo came for Dr. Seuss today, and I am in despair. The Cat in the Hat was my very first book, and I take this personally. "When they came for Mr. Potato Head I said nothing because I wasn't a starchy plant tuber ..."

https://apnews.com/article/dr-seuss-books-racist-images-d8ed18335c03319d72f443594c174513
Do We Need Therapy? Psychology and the Problem of Human Suffering: What Works and What Doesn't?
He spent so much time in bed and felt flat and without any meaning. This seems to me to be the ultimate expression of nihilism. — Jack Cummins

Depression can not be labeled by a philosophical position. It seems to me that being a nihilist could be very freeing; and I can well envision an ecstatic nihilist. Your friend wasn't depressed because he was a nihilist. He was depressed, and coincidentally he may or may not have been a nihilist, or a Presbyterian, or a Druid, or a logical positivist, or a Marxist, Karl or Groucho. None of those intellectual orientations would bear upon a person's depression, which is something else entirely.
Is there a race war underway?
Until we replace autocratic industry with democratic industry ... — Athena

A dictatorship of the proletariat? Am I reading you right?
Is there a race war underway?
Haven't read the thread, just wanted to toss something out that I heard.

The Occupy protests were about class issues. "Banks got bailed out, we got sold out!" was a common refrain. If you look up the frequency of the word "racism" in the news, it was flat for a really long time then started moving up dramatically right around then and it has continued to this day.

The theory being, that the powers that be needed to deflect attention from class issues, so they got everyone worked up about race. You ask the kids on campus what's wrong with society and they sace racism. They never notice the class issues, that the global elite are sucking all of the wealth of the nation and destroying the middle and working classes.

Makes sense to me.
The paradox of Gabriel's horn.
calculus doesn't give us the GH paradox. — InPitzotl

Jeez I'm with @jgill here. This is a standard example from freshman calculus. The integral of 1/x from 1 to infinity is infinite and the integral of 1/x^2 is finite. Or 1/x is square integrable but not integrable if you prefer. Algebraic geometry is a little high-powered in this context, it's not needed. Your judgment is off from letting yourself be trolled by @Metaphysician Undercover.
Is the underlying basis of reality infinite?
deleted
The paradox of Gabriel's horn.
Because it cannot be measured. That's what infinite means. — Metaphysician Undercover

soph·ist
/ˈsäfəst/

* a paid teacher of philosophy and rhetoric in ancient Greece, associated in popular thought with moral skepticism and specious reasoning.
* a person who reasons with clever but fallacious arguments.
The paradox of Gabriel's horn.
If the extension is infinite, the volume cannot be figured. You can only figure the volume by assuming that there is an end, a limit, and this is rounding off. But then you are not figuring the volume of an infinite extension. — Metaphysician Undercover

LOL. Well then how do you know the area under the curve is infinite then?
The paradox of Gabriel's horn.
I see that we have a fundamental difference of opinion concerning the logic of spatial areas. I think that it is illogical to believe that a 3d spatial form with an infinite extension in one dimension could have a finite volume. — Metaphysician Undercover

Well ok. What would you say is the volume of the solid of revolution of $y = \frac{1}{x}$ between 1 and $\infty$ when the curve is revolved around the x-axis? Here's the theory and formula, if you forgot.

https://en.wikipedia.org/wiki/Solid_of_revolution
The economy of thought
I insist: "Martha and Mary are sisters. Marta has two nieces who are not Mary's nieces. It's possible?" — Miguel Hernández

Marta and Maria are nuns in a convent. Or, in American slang, they are African-American women.
The economy of thought
Can someone please summarize in that context the usefulness of excluding number one from the set of prime numbers? — Hrvoje

Haven't I seen this trolled around the Internet on at least two other forums? The answer in that SciAm article is perfectly satisfactory.

In any event 1 isn't prime because

* Excluding it makes the statement of the fundamental theorem of arithmetic simpler. This is the theorem on unique factorization into products of prime powers, such as $15 = 3 \times 5$ or $12 = 2^2 \times3$ . If we included 1 as a prime then $12 = 2^2 \times 3 = 1^2 \times 2^2 \times 3 = 1^3 \times 2^2 \times 3$ etc. So you'd have to say "except for powers of 1" in the statement of the theorem.

The reason behind excluding 1 from the set of primes is that we wouldn't/couldn't have a unique prime factorization for a given number: — TheMadFool

Great minds think alike.

* 1 is a unit in the ring of integers and units aren't prime.

* The ideal generated by 1 is not a prime ideal in the ring of integers because a prime ideal must be a proper ideal, and the ideal generated by 1 is not proper.

Take your pick.
The Dan Barker Paradox
"The best argument against democracy is a five minute conversation with the average voter."

-- Attributed to Winston Churchill, though it's hard to know for sure if he actually said it.
Is this quote true ?
A square circle is impossible. — TheMadFool

I wish people would stop saying that. The unit circle in the taxicab metric is a square. There's a picture of a square circle on this page.

https://en.wikipedia.org/wiki/Taxicab_geometry
Joe Biden: Accelerated Liberal Imperialism
Bombs away baby. This is what got liberals dancing in the streets. And we'll be staying in Afghanistan till the cows come home. Liberals just can't get enough of this stuff ever since Hillary signed on to invading Iraq. Beats the hell out of me, I was a peacenik back in the day and still am. Or as Jimmy Dore noted: Biden bombed Syria before he got anyone covid relief.

Have a look at a leftist truthteller. Jimmy Dore supports Bernie. And peace. You know, just like you used to, before Hillary warped your sense of humanity. "You" meaning anyone who supports what Biden did today and objected when Trump did the same thing in 2018. Like Jen Psaki, who tweeted criticism of Trump's bombing of Syria on the grounds that they're a sovereign country. Guess that's no longer operative.

Now that the bloodthirsty liberals are back in power this is only the beginning.

https://www.youtube.com/watch?v=10w4MhIEr7Q
Is there a logical symbol for 'may include'?
Maybe modal logic can offer some clues. For example there's a "necessarily true" operator, so the negation of that might be what you want.

https://en.wikipedia.org/wiki/Modal_logic
Solutions for Overpopulation
Overpopulation is the easy target here — TheMadFool

Anything that everyone believes is probably wrong. The truth is, the real problem is underpopulation. The fertility rate in the West is below replacement level. And as poor populations achieve modernity and wealth, their fertility goes down. As women get educated, they have fewer children.

As one striking example, look at Social Security in the US. In 1940 there were 159 workers to every retiree. In 2013, there were 2.8. You call that overpopulation? I call it the opposite. There aren't enough people to keep the system afloat. Other developed nations have the exact same problem. There aren't enough new people to support aging populations.

https://www.ssa.gov/history/ratios.html

Here are some links for your reading pleasure, to serve as an antidote to this particular example of a falsehood that everyone believes is true. Disclaimer, I didn't read each of these links nor do I necessarily endorse their authors nor points of view.

https://www.theatlantic.com/ideas/archive/2019/03/underpopulation-problem/585568/

https://www.businessinsider.com/countries-becoming-demographic-time-bombs-2017-8

https://www.catholicworldreport.com/2011/11/01/underpopulation-the-real-problem/

https://medium.com/@kevin2kelly/the-underpopulation-bomb-594425a6df5f

https://prolifeaction.org/2010/overpopulation/

https://lancasteronline.com/opinion/letters_to_editor/underpopulation-is-the-real-issue/article_8915d5da-261e-11e8-889a-afd51d4a4f13.html/

https://www.internetgeography.net/igcse-geography/population-and-settlement-igcse-geography/over-population-and-under-population/

http://geography-groby.weebly.com/uploads/4/3/3/7/43370205/59_courses_and_cons_of_under_population.pdf

ps -- After I posted, this just happened to pop up on my news feed: Male sperm counts are dropping like crazy over the past 40 years. Every single one of those little swimmers is a potential taxpayer.

https://academic.oup.com/humupd/article/23/6/646/4035689
The Hypotenuse Problem (I am confused)
I don't want to drag you back into a dispute. We can agree to disagree. I'll be fine with any compatible definition, proviso the ideas for its proper application are the same. — simeonz

I read the entire Talk thread and have concluded that I no longer have any idea what a Euclidean space is. LOL. However someone in that thread did reference Spivak's monumental Comprehensive Introduction to Differential Geometry, which on page 1 defines Euclidean space as the set of n-tuples of reals with the usual inner product. So again. clearly this is the modern analytic definition, but it apparently sidesteps the subtleties of classical and affine geometry. But my own preference is for the analytic treatment; just as I view an angle as being defined analytically as an arccosine after the cosine has been defined as the real part of the complex exponential, which itself is defined by a differential equation or a power series. There's no longer any geometry involved in the modern definition of angles; although of course one is free to use one's intuition, as we all do.

Exactly, nomenclature or not. Not all philosophical differences translate to definitions and definitions are merely conventions. That is, there is always going to be some contention and heat on the issue, of who establishes the right linguistic terms for mathematics. I am contented to use either, as long as people understand the philosophical distinction and we can talk about that — simeonz

Ok. Spivak is a differential geometer. He wants to associate, or attach, a little copy of $\mathbb R^n$ to every point of a differentiable manifold. This is the viewpoint of modern geometry and in particular general relativity in physics. Or as Einstein said, once he got his theory back from the mathematicians he no longer understood it.

But in this point of view, there's no underlying space at each point that we coordinatize. Rather, there is a copy of Euclidian n-space at every point of some manifold, meaning a set of n-tuples. There's no secret underlying space under the coordinate space.

But you know, you did elevate me to a higher state of confusion. I'd be the first to agree that if we have a plane, it makes no difference where the origin is. But then the coordinate system isn't the plane and never was the plane. The plane is logically prior to the coordinate system. But from the modern point of view, the coordinate system IS the plane. Or at least it's the Euclidean space. So I have definitely become more confused but at a higher level.
The Hypotenuse Problem (I am confused)
I can see that you are being polite. Thanks for not sending me out with a curse. — simeonz

Well, like I say, I'm not entirely sure what we're disagreeing about. And you did actually make me think that I could be missing some subtleties. I know that we can impose a coordinate system on an already-existing object. And I know that Euclidean space is defined (at least by Spivak) as the set of n-tuples itself. So there's some subtle philosophical difference between a coordinate system imposed on an object, versus the coordinate system itself being the object. So I don't think you're entirely wrong. In any case I find the Talk page to that Wiki article interesting, as some of these points are brought out; for example the distinction, or lack of distinction, between $E^n$ and $\mathbb R^n$ . And I'm only rude on this site to people who really really deserve it, and not that often.
The Hypotenuse Problem (I am confused)
No. In fact, it is one of the few places which concurs with the manner in which I was taught to think of analytic geometry. — simeonz

Ok. I'm out of ammo. Maybe you're right.
The Hypotenuse Problem (I am confused)
Many places define it this way. — simeonz

I'm going to gracefully bow out, or turn tail and run, as the case may be. I find myself passionately defending my side of an argument without even knowing what the argument's about.

Let me just refer you to the Talk page for the article in question, https://en.wikipedia.org/wiki/Talk:Euclidean_space . It has many passionate and bitter responses to the article that mirror some of the concerns expressed in this thread. IMO the article itself is a mess. But even so, after waving their hands and confusing the issue massively, and clearly inducing many of the confusions that you've been expressing, they finally give a technical definition:

A Euclidean vector space is a finite-dimensional inner product space over the real numbers.

Which frankly is, on the one hand, at least consistent with my definition as a set of ordered n-tuples with the Euclidean norm; but on the other, is a little messy, because an inner product space is a far more complicated thing than a Euclidean space. Suppose we look up what's an inner product space? We find that, "In mathematics, an inner product space or a Hausdorff pre-Hilbert space[1][2] is a vector space with a binary operation called an inner product." Well that's helpful. If you've studied Hilbert spaces or functional analysis or quantum physics, or know what a Hausdorff space is, and know the difference between a Hilbert space and a pre-Hilbert space, you can maybe figure out what they mean by a Euclidean space.

I pronounce this article hopeless. The Wiki article is trying to blend too many disparate concepts from history and modern practice, trying to be both technical and beginner-friendly, and in the end obscures more than it clarifies. I wonder if you got your ideas just from reading this disaster of an exposition. The Talk page is unusually passionate, as Wiki Talk pages go, in their objection to the content of the main article. You should give it a read. The first paragraph is titled, "Wrong, wrong, wrong," and the rest of the Talk page goes on from there.

Can you at least tell me, did you come by your ideas solely from reading this article?

Let me suggest this. Ignore the Wiki article entirely. A Euclidean space of dimension n is the set of ordered n-tuples of real numbers with the Euclidean norm |x| as I defined it earlier. That definition requires only that you know what an ordered n-tuple of real numbers is. It's accessible to high school students. And from it, you can derive ALL of the properties of Euclidean space including the metric, the inner product, and the vector space and Hilbert space structure. That's the right definition.
The Hypotenuse Problem (I am confused)
How are you detaching from the use of preferred origin and axial orientation then? — simeonz

Detaching from the use of preferred origin and axial orientation? I just can't parse that at all.

Since, obviously, you are defining some point to be (0, 0), — simeonz

Yes, the ordered pair of real numbers (0,0).

and some vectors to be (0, 1), (1, 0). — simeonz

Yes, the ordered pairs of real numbers (0,1) and (1,0), respectively. The ordered pairs ARE Euclidean space, they're not imposed on some underlying space.

I get that you must be making some point about coordinate systems, but your exposition is not adding clarity.

For the universe, (0, 0) would be its center of gravity, — simeonz

I don't know anything about the universe. Other than what Einstein pointed out, that there is no preferred frame of reference. You can put the origin of a coordinate system anywhere you like.

or some other choice that someone deems excellent, for example. — simeonz

Yes. We agree. You can put the origin anywhere that's convenient in any given context.

But in my variant, the choice is made by the use of Cartesian coordinate system which uses orthornormal basis and origin after the fact. The underlying space has no (0, 0) in it, just abstract locations, and there are no special orientations or planes, just abstract vector directions. — simeonz

Ok. I no longer know why I'm even in this thread. May I withdraw gracefully now? I have nothing new to add. Except that I read the OP's initial post (the OP's OP) and I don't think the hypotenuse means anything at all in the graph of apples versus dollars. It's like noticing that your thermometer reads 70 degrees Fahrenheit and that the mercury has reached a height of three inches above the base of the old-fashioned mercury-filled wall thermometer. The three inches is a true measurement, but it has no meaning in the context of measuring heat.

Wikipedia — simeonz

When you copy Wiki paragraphs could you please give the full link? I can't search every Wiki article on coordinate systems, vector calculus, Euclidean space, inner product spaces, and so forth in order to see what the context is.

I don't doubt that such sources are authoritative in their own right. I think that such treatment is a little outdated in style, because the mathematics skip a little modern abstraction, in pre-Russellian (pre-Frege) manner of thought. — simeonz

Jeez, not that Michael Spivak needs the likes of me to defend his reputation. but this remark is a little off target. A lot off target in fact. Calculus on Manifolds is essentially a proof of the generalized Stokes' theorem from the viewpoint of modern differential geometry. It's a very modern book. despite its 1965 publication date. I wonder if you are thinking of something else.

I do not oppose the dot product and metrization you provide. It indeed fits the axiomatic requirements. Affine spaces can be defined over n-tuples (as both point and product spaces) and that Cartesian coordinate systems can simply be rigid transformations over some preferred innate coordinates. However, I have something else in mind. — simeonz

I can see that you do. It's just that you haven't explained it to me. I'll agree with you that we can impose a coordinate system on an arbitrary space, and that the space isn't the coordinate system. But Euclidean space is in fact the coordinate system. Euclidean space is exactly the set of n-tuples with the usual norm, distance, and inner product. That doesn't mean that your ideas about coordinate systems are wrong, it just means that after all this I still can't figure out why we're having this conversation. I should have quit while I was behind a long time ago.

As far as the OP, the length of the hypotenuse doesn't mean anything at all in the context of the graph of the prices of apples.
Complexity in Mathematics
It is a continual curiosity when a person insists on posting opinions on a technical subject of which he or she has not read even the first page in an introductory textbook. — GrandMinnow

Stop picking on @Metaphysician Undercover!
The Hypotenuse Problem (I am confused)
I would probably never deal with underlying vector spaces explicitly. — simeonz

There is no "underlying" vector space. The n-tuples ARE the vector space.
The Hypotenuse Problem (I am confused)
I agree with the fact that we can define the dot product as you specify, but we need inner product as well, or we are just manipulating unitless numbers that don't correspond to anything. — simeonz

That's like saying I'm going to the store for oranges but I need to buy fruit as well. Oranges are the fruit I need to buy. An inner product is an abstraction of the dot product. You can call the dot product the inner product if you like and I usually do. But you are making a distinction that's not really there and introducing confusion. Is this something you got from a book? Maybe this is something I don't know about. You don't have a dot product AND an inner product. You have a dot product which can also be CALLED an inner product. They're the same thing, namely $\displaystyle x \cdot y = \sum_{i = i}^n x_i y_i$ where the $x_i$ 's are the coordinates of $x$ and likewise for the $y$ 's.

But I was saying that there is one more hop (probably) in my mind to how this intuition translates to Cartesian coordinates. — simeonz

That hop is indeed in your mind and you're confusing the issue. The n-tuples ARE the points and the Euclidean norm is ALL the structure you need to define the distance and the dot (or inner) product, which gives you all the structure of Euclidean space.

You're thinking that the n-tuples are imposed on top of an existing space, and perhaps for some purposes that might be a useful point of view, but I don't think it's helpful here.
The Hypotenuse Problem (I am confused)
A Cartesian coordinate system is an assignment of n-tuples to the points in a point space underlying Euclidean space, such that the dot product between the n-tuples is isomorphic to the inner product between the displacement vectors of the points from the origin. — simeonz

Oh my. I think that's hopelessly convoluted, where did you get it?

Here's what Euclidean space is. My reference here is for example Calculus on Manifolds by Spivak, page 1. [That's a pdf link].

Given the real numbers $\mathbb R$ and a positive integer $n$ , Euclidean n-space is defined as the set of n-tuples $(x_i)$ with norm $\displaystyle |x| = \sqrt{\sum_{i=1}^n x_i^2}$ . Spivak writes his indices upstairs ( $x^i$ rather than $x_i$ ) in the manner of differential geometers, but we need not do that here.

The rest of what you wrote is overloaded with what software developers would call cruft. There is no underlying point space, the n-tuples ARE the points. There is no underlying Euclidean space, it's the norm defined on the n-tuples that characterizes Euclidean space. And an inner product is just an abstraction of the dot product, there's no isomorphism going on. It's true that one could in theory define different inner products on Euclidean n-space but I believe (if I recall and I didn't take the trouble to look this up) that they're all related by a linear scaling factor. Or at worst they all induce the same topology via the metric $d(x,y) = |x - y|$ so there's no important difference. I could be wrong, maybe there's some weird inner product you could put on the n-tuples but I don't see how that's important here.

I don't think I should comment on the rest of what you wrote because you have a lot of extra baggage in that one paragraph that's leading to a lot of conceptual confusion. So let's stay here and work this part out.
The Hypotenuse Problem (I am confused)
Those are not analytic. They are intuitional. — simeonz

Well those are not mutually exclusive. Of course we use geometric intuition to get the analytic approach off the ground, but that's true of everything. Is that what you're saying, that we need the ancient geometric intuition to ground the modern analytic approach? Perfectly well agreed. But again, what of it? Nobody's disagreeing.

ps -- I see that you're not the OP. I should quit while I'm behind here. What I know about all this is that inner product spaces are a vast abstraction of ancient Euclidean geometry. But who would disagree? The law of cosines was known to Euclid and is the same concept as the dot product.
The Hypotenuse Problem (I am confused)
I say that the Pythagorean theorem applies to affine spaces over inner product spaces, — simeonz

That sounds right. I don't know much about affine spaces. But basically an affine space is a vector space that's "forgotten its origin" and you don't need any privileged origin to have the Pythagorean theorem be true.

But you are saying this as if someone is denying it. I don't think anyone is denying that the Pythagorean theorem is false in affine spaces. Help me understand what is the point of the thread. I don't think anyone disagrees with what you said here.

I don't see it that way really. We still come from the geometric perspective, to define angles and distances in one way or another, and only then we have the privilege of calling an n-tuple of points being from a Cartesian coordinate system. — simeonz

No that's not true. We define $\mathbb R^2$ as the set of ordered pairs of real numbers. Then we define the usual Euclidean Euclidean distance, and we define the usual dot product. Then the angle between two vectors is the arccosine of the dot product of their normalized versions. That is,

$x \cdot y = |x| |y| \cos \theta$ so that $\theta$ can be defined as $\arccos(\frac{x \cdot y}{|x| |y|})$ . I assume you agree. And we can even formalize the arccosine by defining the cosine as the real part of the complex exponential function, and the arccos as its inverse. All this can be done without reference to geometry and we can even define angles without geometry. I'm guess you know this but disagree for some reason?

Cartesian coordinate systems come with semantics that need to be defined apriori. They are not just mechanical assignment of pairs of numbers to some arbitrary point space. — simeonz

No they don't and yes they are. You just define Euclidean n-space or in general you can define an abstract inner product space and everything works out fine without geometric semantics. For example if instead of Euclidean n-space we can work in generalize inner product spaces and all the theorems carry over directly. I can't see the point of objecting to this but maybe I'm misunderstanding you.

https://en.wikipedia.org/wiki/Inner_product_space

There is only one sense, in fact, in which I am not correct. And it is that a Cartesian coordinate system might be a applied to the very n-tuples, with vectors being n-tuples, distances and angles computed in the usual way, etc. But then, we couldn't talk about apples and dollars, because since the underlying point space is just a mechanical bonding of numbers, it is unitless. — simeonz

There's no Cartesian coordinate system in an inner product space but there is a notion of an orthonormal basis. That's Fourier series, functional analysis, and quantum physics based on Hilbert space. All this is standard. I don't follow your point.

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