Well, in the current theory of the physical world (standard model, or whatever variant of it you prefer) all atoms of the same element are supposed to be EXACTLY the same (indistinguishable, even in principle, with absolute precision), right? — Mephist
You are right, we will never be able to check if this theory is correct with absolute precision, not even in principle, because all physical measurements must necessarily have a limited precision.
Nevertheless, in principle (if you have enough computing power and the model is complete and consistent - I know, that's a big if) you can use the mathematical model to make predictions about the result of experiments with arbitrary precision. — Mephist
So, in a model of the physical world where all distances have to be multiple of a given fixed length (I don't know if such a model exists, but let's assume this as an hypothesis), there cannot be squares
made of unit lengths. — Mephist
I don't know what these unit lengths are made of: they are simply the building blocks of my model, the same as the "strings" of string theory or the "material points" of Newtonian mechanics! — Mephist
By the way, to be clear, I don't believe in this theory! — Mephist
To state the problem succinctly, set theory allows that two distinct things have the same identity, in the same way that we might say two distinct things are equal. The faulty premise is that things with the same value "2" for example, are the very same thing. In other words set theory premises that, "2" refers to an object, rather than a value assigned to an object. It is a category mistake to treat what "2" refers to, as a particular object, rather than as a universal principle. — Metaphysician Undercover
"The square root of two" has no valid meaning in the rational number system. This means that taking a square root is not a valid operation. — Metaphysician Undercover
Now if you want to say, "Yes but you admit numbers aren't real, they're only an abstraction!" I respond: Yes exactly. And traffic lights aren't real either, they're only an abstraction. Law is an abstraction. Government is an abstraction. Science is an abstraction. The Internet is an abstraction. Humans have the power of abstraction. It's how we crawled out of caves and built all this. — fishfry
It's those pesky noncomputable numbers again, one of my favorite topics. — fishfry
You are saying that there are 2 books and two fish and 2 schools of thought; but there is no 2 in the abstract.
Well, imagining or mentally conjuring up a "thing" that is 2, by itself, is one of the greatest intellectual leaps of humanity. As I've noted before, you appear to reject the very concept of abstraction. — fishfry
Well, imagining or mentally conjuring up a "thing" that is 2, by itself, is one of the greatest intellectual leaps of humanity. As I've noted before, you appear to reject the very concept of abstraction.
The invention of the concept of number was a great leap forward for mathematics and also for civilization. — fishfry
That let us study 2 + 2 without having to say 2 fish plus 2 fish and then having to re-calculate 2 elephants plus 2 elephants, and then still not being sure about 2 birds plus 2 birds. — fishfry
t's the power of abstraction that allows us to handle all these cases at once.
You reject abstraction. You're not wrong. It's just a nihilistic philosophy of math and of civilization. Everything about our lives is abstraction. We can't live without abstraction. How do you live without abstraction? How do you function, not believing in numbers? — fishfry
So 3 x 3 = 2. That is, 3 is a number that, when squared, results in 2. So in the integers mod 7, 3 is the square root of 2. Just to startle people I'd go as far as to write — fishfry
Like every statement in math, its truth value depends on the context. In the context of the integers, the statement is false. In the context of the integers mod 7, the statement is true. — fishfry
I think your thesis "stick to finitism when teaching basic math" misses the obvious point of how incredibly messy and complex finitism is, both as a mathematical approach and as a practical application. The overwhelming majority of mathematical applications are based on the continuum - physics, engineering, etc. — SophistiCat
The way I reason about it (ie, as a software engineer), real numbers specify the convergence characteristic of approximation processes that deal with real world problems. What you are saying is that people should study the numerical methods that approximate real world solutions, but shouldn't study analysis of this essential characteristic, which seems to me questionable. Maybe your point relates to the general debate in society - whether engineers should study only constructions and hands-on skills and not analysis (how to derive properties of those constructions), but even then I am leaning towards the usefulness of theoretical understanding. — simeonz
Now those are strange entities, unlike the essentially finite square root of 2 (as you've already noted.) A dark ocean of infinitely informative numbers that can't be named is far more poetic and disturbing than little old 2‾√. — mask
Especially considering the algebraic approach that you presented. I find it intuitively satisfying without considering set theoretic foundations. And the Dedekind cut is satisfying if one can admit sets of rational numbers (intuitively self-supporting, IMO). — mask
I think of them as the "dark matter" of the real number line. We can't name them, we can't compute them, we can't use them for anything. But without them, there aren't enough reals to be Cauchy-complete. — fishfry
There are more things in heaven and Earth, Horatio, / Than are dreamt of in your philosophy -- — fishfry
Please explain this to Metaphysician Undercover! I've had no luck. — fishfry
Actually, yes. Dedekind cuts are another constructive approach. Not too different in spirit, I would say.the reals can be built as cuts or Cauchy sequences. I like cuts for not being equivalence classes. It's an aesthetic preference. Cuts are beautiful ('liquid crystal ladders'). — mask
Rational numbers are actually quite nasty, if you want to work with them in computations. They are pleasant, if you are performing a finite number of arithmetic operations. But assuming a "fraction" representation, once you start evaluating some recursive formula, the numerator and denominator become unmanageable quickly. I am not sure how fast the periodic part in the repeating decimal representation grows, but I wouldn't want to work with that either. That is why software uses only fractions with denominator in exponent form (*) and represents the rest approximately. Correspondingly, algebraic numbers, and even computable transcendentals, are not that bad, when compared to arbitrary fractions.Was it Cantor who said the rational numbers are like the stars in the night sky and the irrationals are like the darkness in the background? Perhaps this has been posted before. — jgill
Actually, yes. Dedekind cuts are another constructive approach. Not too different in spirit, I would say. — simeonz
I am only stating this, because rationals are looked upon so favorably, but I find that their simplicity is somewhat overstated. — simeonz
This is exactly the geometric interpretation that got me into trouble. :) It assumes that rays have points corresponding to every non-negative real number (or lines have points corresponding to all real numbers.) To which, I remember my brain screamed, how do you know? Of course, if this is just analytic geometry, it would be true by construction, but then the argument becomes circular. So, I was asking essentially, how do we know that lines/rays, as they appear in real life, are complete. They could be, or they might not be, but how would a mathematical textbook use something like that, that we know very little about (i.e. space), and which is not axiomatic in nature, and use it to define a mathematical concept. At least for me, it didn't work, and caused me some difficulty.One can construct the positive real numbers as a simpler version of the cuts. In the version I like we have a ray of rational numbers that starts from zero (a subset of Q that is closed downward with no maximum.) I like this for its intuitive connection to magnitude/length. — mask
I understand. After all, this is the rationals' whole gimmick - they are dense. Of course, the finite decimalIMV, the rationals are quite difficult. We know how students hate fractions. But I like the idea of 1/n as a kind of flexible unit. Then m/n is just m of those units. We can adjust n to increase or decrease resolution. And we can do various conversions. So it's difficult but still (after much work and thinking) ultimately intuitive. At least for me. — mask
If you mean electron microscope photos of a lattice of atoms, those are still subject to the quantum and classical measurement problems. To clarify what I said earlier:
* In quantum theory, nothing has an exact position at all. Before it's measured, it doesn't have a position. Sometimes that's expressed by saying that it's in a "superposition" of all possible positions. Then when you measure the particle, it (somehow -- nobody understands this part) acquires a position drawn randomly from a probability distribution.
This applies to all objects, large and small, though the effect is much more pronounced when an object is small.
For example you yourself are where you are in space right now because that's the most likely place for you to be. It is statistically possible that you might suddenly find yourself in a statistically improbable place. For example all the air molecules in your room could move to the corner of the room and you'd have no air. That is extremely unlikely, but it has a nonzero probability. It could happen.
So even if all instances of a given particle are the same, you still have no idea exactly where it is, or exactly how long a line made up of these particles is.
Atoms, by the way, are way too large and they're all different. I don't even know if two hydrogen atoms are exactly the same.
However it's interesting that every electron in the universe is (as far as we know) exactly the same. Why is that? It's another thing nobody understands. — fishfry
* And even in classical physics, a measurement is only an approximation.
So now I'd like to re-ask your question but pertaining to electrons, which are all exactly the same. But electons are very small and extremely subject to quantum effects. You simply can't say exactly where an electron is at any time. Only where it's statistically likely to be. One, because nothing is exactly anywhere at all in quantum physics; and even when it is, after a measurement, the measurement itself is subject to classical approximation error. You made the measurement in a particular lab with a particular apparatus, built and operated by humans. It's imperfect and approximate from the getgo. — fishfry
Well there are no computers with arbitrary precision. That's the problem with the computational theory of the universe. There's too much it can't account for.
It's those pesky noncomputable numbers again, one of my favorite topics. If the universe is "continuous", in the sense that it's modeled by something like the real numbers; then it is most definitely not a computer or an algorithm. Because algorithms can't generate noncomputable numbers. — fishfry
So in your hypothetical world there would be squares and if you want to go from (0,0) to (1,1) you simply have to move 2 units, one unit right and one unit up. You can't travel along the diagonal because at the finest level of the lattice, you can't move diagonally. I have no idea what that means physically but I think you are overthinking this or underthinking it. It's kind of tricky, which is a problem for the theory. — fishfry
Some people do! There are some discrete or quantized theories of reality around, like loop quantum gravity. From the article: "The structure of space prefers an extremely fine fabric or network woven of finite loops." — fishfry
But I don't speculate about the physical world. Math is so much simpler because it doesn't have to conform to experiment! In math if you want a square root of 2, you have your choice of mathematically rigorous ways of cooking up such a thing. — fishfry
I will allow myself to interject, although the physics involved in your discussion appears beyond my competence. In any case, just because something is not physical, doesn't make it purely fictitious.I agree with you on the square root of 2, of course! But I am not so convinced that mathematical objects are only cooked-up fictions not related to physical reality. — Mephist
This is exactly the geometric interpretation that got me into trouble. :) It assumes that rays have points corresponding to every non-negative real number (or lines have points corresponding to all real numbers.) To which, I remember my brain screamed, how do you know? — simeonz
I started to question what was the goal - were numbers lexical entities or geometric properties? What was that we were trying to define - quantities, computations, geometrical facts? How could we validate them? At this point I didn't know much about algebraic structures and axiomatic systems. I went completely on a pseudo-philosophical tangent and refused to learn anything whose methodological grounds I did not understand completely. — simeonz
They could be, or they might not be, but how would a mathematical textbook use something like that, that we know very little about (i.e. space), and which is not axiomatic in nature, and use it to define a mathematical concept. At least for me, it didn't work, and caused me some difficulty. — simeonz
I know you don't like technical stuff by I'm pointing out that I don't need set theory to build a square root of 2. — fishfry
Especially considering the algebraic approach that you presented. I find it intuitively satisfying without considering set theoretic foundations. And the Dedekind cut is satisfying if one can admit sets of rational numbers (intuitively self-supporting, IMO). — mask
Alternatively, the notion of a real number from abstract algebra is one of a complete ordered field. Ultimately, it is the same concept. The properties are the same, except that the approach to the investigation is leaning more heavily towards non-constructivism. Which is fine, because this is what abstract algebra is all about. In fact, in some sense, the abstract definition is the proper definition, and the constructive one serves as an illustration. The latter is pedagogically necessary, but once understood, is not essential anymore. — simeonz
Nevertheless, how you stipulate or construct the object lends a particular perspective on what it means; even when all the stipulations or constructions are formally equivalent. — fdrake
The problem of irrational numbers arose from the construction of spatial figures. That indicates a problem with our understanding of the nature of spatial extension. So I suggested a more "real" way of looking at spatial extension, one which incorporates activity, therefore time, into spatial representations. Consider that Einsteinian relativity is already inconsistent with Euclidian geometry. If parallel lines are not really "parallel lines", then a right angle is not really a "right angle", and the square root of two is simply a faulty concept. — Metaphysician Undercover
The operation on the group was 'really' functional composition, which is why groups weren't automatically commutative. — mask
For someone who insists on math being beautiful, it has to sing for the intuition. For example, when learning group theory I really liked thinking of groups of permutations. Those were the anchor for my intuition — mask
Every philosophy is tinged with the colouring of some secret imaginative background, which never emerges explicitly into its train of reasoning.
Get involved in philosophical discussions about knowledge, truth, language, consciousness, science, politics, religion, logic and mathematics, art, history, and lots more. No ads, no clutter, and very little agreement — just fascinating conversations.