But perhaps someone reading this took a course in groups, rings, and fields but forgot this beautiful construction, which we can sum up in one equation: — fishfry

Thereafter I avoided abstract algebra. — jgill

I think we agree that it's bad pedagogy to simply posit the reals with no explanation and no time or ability to answer very expected and natural questions. — boethius

Seems that way to me too. Our friend Metaphysician Undercover, who must be a neo-Pythagorean, is mightily vexed by the fact that the square root of 2 is (a) a commonplace geometric object, being the diagonal of a unit square; and (b) doesn't happen to be the ratio of any two integers.

What of it? Humans got over this about 2500 years ago. — fishfry

It was quite a shock to them when they came to realise that the square root of two could not be expressed as a fraction! There was no alternative except to introduce real numbers. — A Seagull

So, my challenge is if someone can construct the real numbers in a concise and clear way that the average student starting calculus in high school would easily understand for then transcendental constants like pi to make perfect and clear sense and all the tricky questions above perfectly clear answers (just as clear as in geometry or proofs about discrete numbers). — boethius

Human beings may have gotten over this, but they did not resolve the problem — Metaphysician Undercover

Human beings may have gotten over this, but they did not resolve the problem. — Metaphysician Undercover

Consider the problem this way. Take a supposed "point". Now measure a specific distance in one direction, and the same distance in a direction ninety degrees to the first. Despite the fact that you use the exact same scale of measurement, in both of these measurements, the two measurements are incommensurable. Why is that the case? — Metaphysician Undercover

Doesn't this tell you something about the thing being measured (space)? — Metaphysician Undercover

What it tells me, is that this thing being measured (space), cannot actually be measured in this way. The irrational nature of pi tells us the very same thing. Two dimensional objects have a fundamental problem which demonstrates that space cannot actually be represented in this way. — Metaphysician Undercover

We see a very similar problem in the relation between zero dimensional figures (points), and one dimensional figures (lines), as discussed in the other thread. So if we get done to the basics, remove dimensionality and focus solely on numbers, we can learn to understand first the properties of numbers, quantity, and order, without applying any relations to spatial features. Then we can see that it is only when we apply numbers to our dimensional concepts of space, that these problems occur. The problems result in establishing a variety of different number systems mentioned in this thread. None of these numbers systems has resolved the problem because the problem lies within the way that we model space, not within any number system. We do not have a representation of space which is compatible with numbers. — Metaphysician Undercover

Two dimensional objects have a fundamental problem which demonstrates that space cannot actually be represented in this way. — Metaphysician Undercover

I'm not building with infinite integers, I'm building with the infinite decimal expansion representation of real numbers and simply pruning off the decimal symbol. Sure, if we simply define integer as "not this" then it's not building an infinite integer, but it is building something that I can then do things with if I'm not prevented from doing so. — boethius

However, if we switch regimes to one where we now have access to the infinite digit expansion of real numbers, we can revisit every proof in the previous regime with our new objects; and now, revisiting the root 2 proof is irrational I am able to solve it with these new objects and not arrive at a contradiction as oddness / eveness is no longer defined upon which the classic proof by contradiction depends. This is what I am doing. — boethius

Am I prevented from doing this full stop? Am I unable to find a "suitable decimal expansion" to solve my problem? What exactly is preventing me from doing this, that is what I would consider a suitable answer in the context of learning maths. Given these infinite decimal expansion, I want to use them as what ways I see fit, unless I'm prevented by some axiom — boethius

Broad features and themes involved in rigorous proofs elsewhere I do not consider a good answer for learning math. For me, "learning math is" understanding the proof oneself, not understanding that others elsewhere have understood something. — boethius

Moreover, your approach, would seem to me, to imply that the decimal expansion representation of a real number cannot be counted; is this your implication? or would you say the digits in a real number are countable? — boethius

Also, how do you maintain infinite sequences can be completed, there are no infinite integers, the sequence of integers is infinite, simultaneously within the system suitable for high school level maths. Do we simply elect not to use our "complete the infinity tool" on the integers, and add this axiomatically? — boethius

What axioms do they have to work with? Do they know enough set theory do make a model that avoids all these problems, or do they have another suitable basis? — boethius

I think we agree that it's bad pedagogy to simply posit the reals with no explanation and no time or ability to answer very expected and natural questions. Instead of curiosity leading to better understanding, it leads to confusion and a sense maths is "because we say so", which is the exact opposite sense students should be getting. — boethius

Students would be better served by a less ambitious (not actually having irrationals and transcendentals as objects) but more rigorous calculus in the numerical regime, which would make a much more solid foundation for students going on to use applied maths, who can simply stay in this regime (as they will likely be solving every problem with the computer), and better serving pure maths students as well (that mathematics is rigorous, and extensions are made to do new things in a rigorous way). — boethius

The digits in a real number should not be countable, but you have to say which algorithm you use to generate them, since they are infinite. — Mephist

I would say:
- infinite sequences are the same thing as functions from integers to sequence elements.
- functions from integers to sequence elements are surely well defined if the rule to produce the Nth element is clear (is an algorithm)
( maybe explain that you can even assume the existence of non-algorithmic functions, with the axiom of choice, but you cannot use it freely without making use of formal logic )
- integers are defined as sums of powers of 10 (that is the DEFINITION of an integer in the standard notation, not some strange property. So, 2 is 1 + 1 BY DEFINITION: nothing to be proved). The problem with infinite integers is that you don't know which powers of 10 it's made of. If you have an infinite decimal expansion, you know the powers of 10 and everything works. If you are not convinced, try to write infinite integers in Peano notation: 1+1+1+1+.... (or SSSSS..0 - same thing): they are all the same number.
- the sequence of integers is infinite because is constructed by adding +1 at each step, and this is a non terminating algorithm that produces a well defined result at each step, so it's allowed as an algorithm. — Mephist

Well, I think a lot of interesting calculus at Euler's level could be done in a enough rigorous way, and just make the students aware of what are the really rigorous parts and which ones are the most "doubtful", when reasoning about infinities. But the most doubtful ones are even the most interesting! — Mephist

What for should I (as a student) loose time in a subject where everything hast just been discovered long time ago, and the only thing I can do is to learn by mind what others did? Math becomes interesting when you see that you can use it do discover new things that nobody said you. And there are still a lot of things to be discovered; only that you have to learn how to reason about them in the right way! — Mephist

However, the other problem I've been alluding to is revisiting all previous theorems proven in a finitist regime; which is also essential part of understanding the infinite regime. Some theorems are abandoned. Choices must be made.

For instance, in Euclidean geometry we can have a theorem that sphere represented by an arbitrary amount of components, but not infinite, cannot be turned into 2 equal spheres of equal volume. We can also have a theorem that arbitrary amounts of lines never completely fill up area or volume. Going to the standard infinite regime we can revisit these theorems and prove them "false" in the sense that what we thought we couldn't do before we can do now in this new system. This, for me, these "side-affects" features that we didn't expect and didn't set out to make, is what makes these areas of mathematics difficult, even more than being able to construct objects we're intending to make like the real numbers, and high school students. — boethius

So, even if there was time to explain infinite digit expansion is uncountable in some actual mathematical way involving definitions and proofs, and it's due to this uncountability that's we can assert they cannot be converted into integers ... while still having infinite integers but no "infinite integer" available to put in our set of rationals ... neither asserting that all integers in our set are finite in a sense of having a finite amount of them, which would be clearly false. — boethius

So, infinite sets and real numbers could be something introduced at the very end of secondary maths when these foundational concepts are more familiar. But to start, understanding divergence and convergence and tangents and how series and sums and derivatives and integral functions relate to each other (and how to solve real problems with them), is challenging enough to learn in a finitist regime; my intuition is that doing this also with the conceptual challenge of infinity makes it much harder to "see" and to "get" what's going on, and students who start asking questions, even just pondering to themselves, that have no good answers available will much more easily get lost or believe their questions are seen as stupid by the mathematical community, simply because their teacher can't deal with them. — boethius

What about inverses? It's not immediately obvious, but in fact if
$a+b\sqrt{2}\ne 0$ then $\frac{1}{a+b\sqrt{2}}=\frac{a}{{{a}^{2}}-2{{b}^{2}}}+\frac{-b}{{{a}^{2}}-2{{b}^{2}}}$. — fishfry

OK, then there is this little "glitch" in the fabric of the universe named Banach Tarski theorem... :smile: — Mephist

It doesn't work because in integral calculus you have to take "open sets" as infinitesimal pieces ( but I would prefer to not go into details about this issue, because surely fishfry will read this and will not agree :wink: ) — Mephist

What's wrong with the Banach-Tarsky paradox — Mephist

But you did give a wrong and misleading definition of an open set. I do have to say that. Open sets are really important. An open set in the reals is just like an interval without its endpoints. What matters about it is that "all its points are interior points." It doesn't include any points of its boundary. That's what makes open sets have the interesting properties that they do.

They're not really infinitesimal. They can be arbitrarily small. But they aren't "infinitely" small. In fact that is the great "arithmetization of analysis," the great founding of the continuous world of calculus on the discrete world of set theory. Instead of saying things are infinitely small, from now on say they're arbitrarily small. For every epsilon you can go even smaller. But in any individual instance, still nonzero. That's the essence of open sets. — fishfry

I wrote "The digits in a real number should not be countable". Well, the digits of (the decimal representation of) a real number are countable, since they are determined by a function of type "natural-number ==> digit". — Mephist

Well, I wouldn't start from the "pathological" cases to show that volume additivity doesn't work any more. — Mephist

The opposite argument is that it's bad pedagogy to expect high school students to understand the sophisticated constructions of higher math. It's true in all disciplines that at each level of study we tell lies that we then correct with more sophisticated lies later. It's easy to say we should present set theory and a rigorous account of the reals to mathematically talented high school students. It's much less clear what we should do with the average ones. Probably just do things the way we do them now. — fishfry

However, with ubiquitous and incredibly powerful computing and no need for physicists to believe in a physical continuum, I would argue the average student is much better served by focusing on "what can the computer do for me", viewing constants algorithmically with arbitrary (to a physical limit of computation) precision potential determined in practice by one's problem, and building up intuitions around machine calculation (and analytical work including error bounds, computational complexity, along with analytical proofs of convergence when available, just in the "arbitrarily close to the limit" finitist framework); rather than, what we seem to all agree here, building up wrong intuitions about the real number system. — boethius

I would say that it works even if you consider infinitesimals as really existing entities — Mephist

As I wrote in my explanation about Banach-Tarski mounts ago, the theorem works because it uses isometric transformations, but applied to set of points that are isolated from each other (not on open sets). If you impose the restriction that your isometric transformations should be even continuous (going from open sets to open sets), you can't do it any more. — Mephist

But your stance here is literally pre-Pythagorean. The Pythagoreans threw some guy overboard for making the discovery, but they accepted the fact of the irrationality of 2–√2. You refuse even to do that. You're entitled to your own ideas, but to me that is philosophical nihilism. To reject literally everything about the modern world that stems from the Pythagorean theorem. You must either live in a cave; or else not live at all according to your beliefs. You must reject all of modern physics, all of modern science and technology. You can't use a computer or any digital media. You are back to the stone age without the use of simple algebraic numbers like 2–√2. — fishfry

No of course not. It tells me something interesting about abstract, idealized mathematical space. It tells me nothing about actual space in the world. — fishfry

There are no irrational distances in physical space for the simple reason that there are no exact distances at all that we can measure. So it's not meaningful to talk about them except in idealized terms. — fishfry

For all these reasons, 2–√2 is essentially a finite mathematical object. You're simply wrong that it "introduces infinity," because you have only seen some bad high school teaching about the real numbers. Decimal representation is only one of many ways to characterize 2–√2, and all the other ways are perfectly finite. — fishfry

But to simply say that you don't like 2–√2; that's just a hopelessly naive viewpoint. — fishfry

It's just a representation, imperfect from the start. — fishfry

I'm not sure what you mean. Can you please link your earlier post on B-T? — fishfry

Mathematics (what is called mathematics today) is the research of "models' factorizations" that are able to compress the information content of other models (physical or purely logical ones). — Mephist

I am satisfied with this principle if we can apply it consistently. We do not measure mathematical "objects", they are tools by which we measure objects. That's why I argued that they are not proper "objects". — Metaphysician Undercover

Now let's apply this to set theory. Cardinality, for example is a measure. — Metaphysician Undercover

If the applicable principle is that we do not measure mathematical "objects", then why allow this in set theory? It's inconsistency. — Metaphysician Undercover

So either we can measure mathematical objects, like squares, and the sides of squares, just like we can measure the cardinality of sets, or we cannot measure these so-called mathematical objects. — Metaphysician Undercover

But if we allow that we can measure these so-called objects, then we can measure a square, and find that the diagonal cannot be measured. — Metaphysician Undercover

It's what we call an "irrational number", implying an immeasurable length. Are you familiar with basic geometry? — Metaphysician Undercover

This is not at all what I've been saying, so I think we might not really be making any progress. — Metaphysician Undercover

Neither you nor I is talking about physical objects here. What we are talking about is the "made-up gadgets" which you describe here. — Metaphysician Undercover

You seem to imply that there is a difference between these funny gadgets, and "first-rate mathematical objects" — Metaphysician Undercover

I deny such a difference, claiming all mathematics consists of made-up gadgets, and there is no such thing as mathematical objects. — Metaphysician Undercover

But this is contrary to set theory which is based in the assumption of mathematical objects. — Metaphysician Undercover

If you really think that a "funny gadget" becomes a "mathematical object" through use, you'd have to demonstrate this process to me, to convince me that this is true. — Metaphysician Undercover

How can you not see that this is a problem for set theory? — Metaphysician Undercover

Set theory assumes that it is dealing with real, actual mathematical "objects". — Metaphysician Undercover

That is a fundamental premise. — Metaphysician Undercover

Now you agree with me, that mathematics can never give us this, real or actual things being represented by the symbols. — Metaphysician Undercover

So why don't you see that set theory is completely misguided? — Metaphysician Undercover

So your argument is that the "funny gadget" gets made into a "first-rate mathematical object" through convention, just like driving laws. — Metaphysician Undercover

But those are ";laws", not "objects". Let's suppose that the mathematical symbols referred to conventional laws instead of "objects", as this is what is implied by your statement. How would this affect set theory? Remember what I argued earlier in the thread, sometimes when a symbol like "2" or "3" is used, a different law is referred to, depending on the context. — Metaphysician Undercover

I don't see how "the square root of 2 exists" could possibly be true, It is an irrational ratio which has never been resolved, just like pi. — Metaphysician Undercover

How can you assert that the solution to a problem which has not yet been resolved, "exists"? — Metaphysician Undercover

Isn't this just like saying that the highest number exists? — Metaphysician Undercover

But we know that there is not a highest number, we define "number" that way. — Metaphysician Undercover

Likewise, we know that pi, and the square root of two, will never be resolved, — Metaphysician Undercover

#!/usr/bin/python3

low  = 1
high = 2

loops = 1000

for i in range(loops) :
    lowsq  = low * low
    highsq = high * high

    trial   = (low + high) / 2
    trialsq = trial * trial

    if trialsq < 2 :  # too small
        low = trial

    else :   # too big. 
        high = trial
    
print(trial)    

I linked it in the post just before this one. Here's the link:
https://thephilosophyforum.com/discussion/comment/302364 — Mephist

Now back to your quote. Of course we measure mathemtical objects. A unit square has side 1, area 1, and its diagonal ... well you know what its diagonal is. In fact it falls out of the Euclidean distance formula as the distance between the origin (0,0)(0,0)and (1,1)(1,1). — fishfry

No, I'm pointing out that there only seems to be a difference depending on what age one lives in. If you live in the age of integers you don't believe in rationals. You're stuck in the age of rationals and don't believe in irrationals. Matter of history and psychology.

…

We're in agreement then. But that's what a mathematical object is. A made-up gadget that, by virtue of repetition, gains mindshare. — fishfry

I've studied set theory and read a number of set theory texts. I've never read or heard of any such thing. Set theory in fact is the study of whatever obeys the axioms for sets. If you ask a set theorist what a set is, they'll say they have no idea; only that it's something that obeys whatever axiom system you're studying.

You're making stuff up to fill in gaps in your mathematical knowledge. Set theory doesn't assume anything at all. It doesn't assume it's "about" anything other than sets; which are things that obey some collections of set-theoretic axioms. — fishfry

But I already did. From the naturals to the integers to the rationals to the reals to the complex numbers to the quaternions and beyond. At each stage people didn't believe in the new kinds of number and though it was only a kind of "calculating device." Then over time the funny numbers became accepted. This is a very well known aspect of math history. — fishfry

But do you mean how can sets be used to model mathematical objects like numbers, functions, matrices, topological spaces, and the like? Easy. We can model the natural numbers in set theory via the axiom of infinity. Then we make the integers out of the natural, the rationals out of pairs of integers, the reals out of Cauchy sequences of reals, the complex numbers out of pairs of reals, and so forth. If you grant me the empty set and the rules of ZF I can build up the whole thing one step at a time. — fishfry

You haven't made any such case. — fishfry

No, I took pains to make a distinction. Driving laws are completely arbitrary. But many mathematical ideas are forced on us somehow, such as the fact that 5 is prime. — fishfry

Here's an example. Take 1/3 = .333333.... Would you say that 1/3 is not resolved or requires an infinite amount of information? But it doesn't. I could just as easily say, "A decimal point followed by all 3's." That completely characterizes the decimal representation of 1/3. I don't have to physically be able to carry out the entire computation. It's sufficient that I can produce, via an algorithm, as many decimal digits as you challenge me to. — fishfry

It has been completely resolved. — fishfry

It has been completely resolved. You can't write down infinitely many digits any more than you can write down all the digits of .333... But in the case of 1/3, there's a finite-length description that tells you how to get as many digits as you want. And with square root of 2, there is ALSO such a finite-length description. Would you like me to post one? — fishfry

I really hope you'll consider the example of 1/3 and the fact that we can predict or determine every single one of its decimal digits with a FINITE description, even though there are infinitely many digits. Square root of 2 and pi are exactly the same. They are computable real numbers. There is a finite-length Turing machine that cranks out their digits. — fishfry

It's just a representation, imperfect from the start. — fishfry