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Stream: theory: philosophy

Topic: Ontological and epistemological status of real numbers

Madeleine Birchfield (Aug 02 2024 at 17:22):

split off from #community: discussion > ACT pedagogy and hype (cross post)

Madeleine Birchfield (Aug 02 2024 at 17:28):

A question I have for everybody. What exactly are the real numbers?

There are so many different models of real numbers, such as Cauchy real numbers, Escardo-Simpson real numbers, Dedekind real numbers, MacNeille real numbers, the locale of real numbers, hyperreal numbers and other non-standard real numbers, Archimedean ordered Kock fields from synthetic differential geometry, Cauchy sequences of rational numbers from some parts of constructive mathematics, rational numbers, dyadic rationals, finite decimals, or floating-point numbers, most of which are provably not equivalent to each other.

Graham Manuell (Aug 02 2024 at 19:17):

I think there is really only one answer, and that is the locale of real numbers (and so if you really want to consider the points, the Dedekind reals). They are the only ones that behave in a reasonable way imo. I don't understand how rational numbers of anything similar could possibly work, since they have the entirely wrong topology.

Graham Manuell (Aug 02 2024 at 19:21):

Kevin Carlson said:

Graham Manuell said:

I guess wouldn't call what computers do to model real numbers the 'real world'. Computers might use fixed point or floating point numbers, but the real world generally does not. Sure, I agree that it is good for people to prove things about how well these models correspond to the real world, but it feels backwards to take these models as the fundamental notion.

What exactly is this "real world" for you that computers are not part of? It seems very intuitively clear to me that any kind of numbers, insofar as they're actually going to be used to model specific concrete phenomena (whether physical, social, whatever) are going to factor through some finite approximation of the real numbers.

Obviously computers are part of the real world, but this point seems somewhat obtuse, since by this standard anything that mathematicians do is also part of the real world. I have the opposite intuition to you regarding finite approximations. I don't know why you say that when you actually model phenomena you use finite approximations, because essentially all of physics does not do this.

Graham Manuell (Aug 02 2024 at 19:24):

I guess the point behind both of my responses is that we already have an entire field of mathematics that allows us to connect idealised 'infinite' objects such as the reals to finite approximations thereof -- namely, topology.

Kevin Carlson (Aug 02 2024 at 19:34):

Yes, I think I said “model” when I should have said something more like “measure”. I mean that any specific instance of an actual measurement or output of an actual calculation has to be in some finite space.

Kevin Carlson (Aug 02 2024 at 19:35):

So, yes, of course physics models the universe as a real manifold, but I certainly wouldn’t confused $\mathbb R^4$ with “the real world”.

Graham Manuell (Aug 02 2024 at 19:50):

Kevin Carlson said:

Yes, I think I said “model” when I should have said something more like “measure”. I mean that any specific instance of an actual measurement or output of an actual calculation has to be in some finite space.

This is true, but using real numbers already allows us to handle this. Using $\mathbb{Z}[1/10]$ would not be appropriate.

Graham Manuell (Aug 02 2024 at 19:54):

It's not obvious to me whether the universal 'actually is' a manifold or not, but this model is certainly closer to reality than floating point numbers. This is what I was pushing back against. (I wasn't the one to introduce the real world terminology and don't really care to defend it, but I do think you know we mean by it.)

Notification Bot (Aug 02 2024 at 20:09):

A message was moved here from #community: discussion > ACT pedagogy and hype (cross post) by John Baez.

Madeleine Birchfield (Aug 02 2024 at 20:32):

Graham Manuell said:

I think there is really only one answer, and that is the locale of real numbers (and so if you really want to consider the points, the Dedekind reals). They are the only ones that behave in a reasonable way imo. I don't understand how rational numbers of anything similar could possibly work, since they have the entirely wrong topology.

What do you say to the predicative constructivists and type theorists who don't believe in the Dedekind reals or the locale of real numbers in their foundations for whatever reason?

Kevin Carlson (Aug 02 2024 at 20:34):

OK, so it's probably clear that floating point numbers are worse than real numbers because reality seems to have some property that you can zoom in arbitrarily far or something. But a structure where you can take measurements valued in arbitrarily fine finite meshes looks like an inverse limit--say an inverse limit of rulers with twice as fine a resolution at every stage, with the transition maps rounding down. This inverse limit will be basically a Cantor space, viewed as the space of actual decimal expansions--$0.\bar 9$ doesn't get identified with $1$, there's just nothing in between them. This distinction comes because the inverse limit constructs the space of possible sequences of measurements, each refining the previous one. So you could actually get the sequence $0.9,0.99,0.999,\ldots$ of measurements on your family of rulers, or you could get the sequence $1.0,1.00,1.000,\ldots$, and since the sequences aren't the same they end up different in the limits.

Kevin Carlson (Aug 02 2024 at 20:34):

While there are lots of mathematical and pragmatic reasons to prefer to take the quotient of this Cantor space and get the usual real line, I'm very curious whether there's anything else--is there any good reason to think that models using the reals rather than the Cantor space is somehow actually "more correct" about how the world is?

Madeleine Birchfield (Aug 02 2024 at 20:44):

Kevin Carlson said:

While there are lots of mathematical and pragmatic reasons to prefer to take the quotient of this Cantor space and get the usual real line, I'm very curious whether there's anything else--is there any good reason to think that models using the reals rather than the Cantor space is somehow actually "more correct" about how the world is?

In constructive mathematics, taking the quotient doesn't preserve Cauchy completeness of the pseudometric space, so one will still have to construct the Cauchy completion of the resulting metric space to get a more well-behaved real line with well-defined analytic functions. Or one can just not take the quotient set and keep Cauchy completeness, but have to deal with the fact that the space is only a setoid.

Graham Manuell (Aug 03 2024 at 10:40):

Madeleine Birchfield said:

Graham Manuell said:

I think there is really only one answer, and that is the locale of real numbers (and so if you really want to consider the points, the Dedekind reals). They are the only ones that behave in a reasonable way imo. I don't understand how rational numbers of anything similar could possibly work, since they have the entirely wrong topology.

What do you say to the predicative constructivists and type theorists who don't believe in the Dedekind reals or the locale of real numbers in their foundations for whatever reason?

They should just work with the locale of reals via a presentation. This is essentially what formal topology is doing.

Graham Manuell (Aug 03 2024 at 10:44):

Kevin Carlson said:

While there are lots of mathematical and pragmatic reasons to prefer to take the quotient of this Cantor space and get the usual real line, I'm very curious whether there's anything else--is there any good reason to think that models using the reals rather than the Cantor space is somehow actually "more correct" about how the world is?

As I said above for the rationals, Cantor space is clearly wrong because it has the entirely wrong topology. If you tried to use Cantor space to model reals you would be able to decide whether a given number is < 1 or not. This is not something that should be possible with measurements of finite precision.

Kevin Carlson (Aug 03 2024 at 19:31):

Graham Manuell said:

Kevin Carlson said:

While there are lots of mathematical and pragmatic reasons to prefer to take the quotient of this Cantor space and get the usual real line, I'm very curious whether there's anything else--is there any good reason to think that models using the reals rather than the Cantor space is somehow actually "more correct" about how the world is?

As I said above for the rationals, Cantor space is clearly wrong because it has the entirely wrong topology. If you tried to use Cantor space to model reals you would be able to decide whether a given number is < 1 or not. This is not something that should be possible with measurements of finite precision.

Ok, sure, I follow, but the thing that’s been irking me lately is this: imagine a digital stopwatch. I measure something, I click it, I get 0.99 seconds. You’re picturing this as meaning the metaphysically correct measurement 0.99+/-$\varepsilon$ for some random variable $\varepsilon$. But maybe I don’t want to make metaphysical assumptions about the existence of an objectively “correct” time I’m approximating, something by definition there’s no way to observe. I just want to talk about digits displayed on stopwatches. There’s no nonconstant continuous map from the real line to the space of digits that can display on a stopwatch, so the topology of the real line is not, I think, good for this purpose. If I want a big space of ideal measurements, I can instead imagine families of synchronized stopwatches with unboundedly many digits, and that gives the Cantor space. It’s true of course that you can’t actually synchronize infinitely many stopwatches perfectly, but you can’t actually observe an exact real number either, both are inevitably idealizations; the point I’m getting at is that it seems valuable to me to note that there is an ideal space on which physical measurements are actually continuous, unlike the reals. To be clear this isn’t at all saying I imagine Cantor space could replace the reals in physical modeling, just wondering about whether it might have a role around these issues of measurements.

Nathaniel Virgo (Aug 04 2024 at 00:34):

I don't know the answer to this question: is that Cantor space invariant to the scale used to measure time? Like if my stopwatch measures time in lunar months instead of seconds, are we measuring the same thing? (I assume the ratio between seconds and lunar months is irrational.)

John Baez (Aug 04 2024 at 07:51):

Kevin wrote:

There’s no nonconstant continuous map from the real line to the space of digits that can display on a stopwatch, so the topology of the real line is not, I think, good for this purpose.

Nature solves this problem by not having the stopwatch display digits at every moment in time. Between the times when it reads 0.99 seconds and the times when it reads 1.00 seconds, there are times when the display is in the process of changing.

How long is the interval when the display is not showing a well-defined number? I tried reading about the refresh rate of a digital stopwatch, but it's hard to get good information about this question. Anyway: there are physical and technological limits on how short you can make it.

John Baez (Aug 04 2024 at 07:54):

Now, you might say you don't care about these real-world limitations because you're talking about math. But you're talking about a stopwatch, so these real-world limitations matter - unless you're talking about some sort of mathematician's idealized stopwatch that can have arbitrary but finite precision.

If we're going to talk about a mathematical idealized stopwatch, we can even imagine one that has infinitely many digits and shows time as an exact real number.

If you prefer talking about an imaginary stopwatches that can have a trillion digits, or a googol digits, but not infinitely many digits - okay, fine, that's a legitimate choice. But I don't see this as dramatically more realistic.

Notification Bot (Aug 04 2024 at 07:59):

This topic was moved here from #community: discussion > Ontological and epistemological status of real numbers by Matteo Capucci (he/him).

Matteo Capucci (he/him) (Aug 04 2024 at 08:00):

(feel free to move this to #theory: mathematics if you feel it's more appropriate)

Kevin Carlson (Aug 04 2024 at 08:28):

John Baez said:

Nature solves this problem by not having the stopwatch display digits at every moment in time. Between the times when it reads 0.99 seconds and the times when it reads 1.00 seconds, there are times when the display is in the process of changing.

Interesting point but I’m not really convinced—surely if you click a stopwatch during the refresh it just reads the number it was working on generating? If you really really zoom in then I guess there’s some electrical signal changing state which is probably happening continuously, but then if you zoom in further it’s all some kind of quantum fuzz anyway and I don’t think that resolves the question. It seems to me that it’s still the case that “clicking a stopwatch is a map into the space of possible screen states” is a very good model of how a real stopwatch works, and so I still have the desire to have that be a continuous mapping from some space of ideal measurements or states.

Kevin Carlson (Aug 04 2024 at 08:31):

Oh, and if we do try to imagine your infinite-digit stopwatch, I’m pretty sure it has a Cantor space of states, not a real line of them!

John Baez (Aug 04 2024 at 08:33):

I could keep arguing, but I really feel you folks living in the universe of digital technology tend to be a bit fooled by it into thinking the universe is digital. You're making me want to buy a good old old fashioned analogue watch.

John Baez (Aug 04 2024 at 08:38):

Digital technology works extremely hard to make a tiny portion of the universe act like it's discrete, digital, deterministic, non-quantum and so on. And it does a good job! But it's like a thin veneer.

John Baez (Aug 04 2024 at 09:46):

While I dislike conversations of this kind (so should have stayed quiet), I do find it mildly interesting to think about how nature accomplishes a mapping from a continuum of states (like a manifold or a projectivized Hilbert space) to a finite set of "meter readings".

One way is that the finite set of meter readings is really a finite collection of disjoint open sets whose union is not the whole set of meter states: there are also other states where the meter doesn't have a well-defined reading. For example if you are looking at a digital watch, there are short stretches of time when it doesn't show any number at all. I mentioned that before.

Another way uses randomness. You can have a nonconstant stochastic map from $\mathbb{R}$ (for example) to a finite set. This covers the sort of randomness that appears in classical mechanics. There is also a quantum analogue of this phenomenon. For example, there's a stochastic map from the space of pure states of a spin-1/2 particle, $\mathbb{C}\mathrm{P}^1$, to the 2-element set $$\{\up, \down\}$$, which gives the probability of measuring the particle to have spin up or down in any state.

John Baez (Aug 04 2024 at 09:53):

When you press a button on a digital stopwatch and get a time reading, this stochasticity comes into play. Your soft thumb slowly exerts more and more pressure on the button, and there is some randomness involved in how you move your thumb, and when the button is pressed enough to transmit a signal to the digital machinery in the watch. This randomness would be significant if your digital stopwatch measures time in thousandths of a second. To increase the accuracy, we'd need to use something other than your thumb to trigger the watch. But no matter what you do, some randomness is unavoidable.

John Baez (Aug 04 2024 at 10:00):

Recently physicists have experimentally measured the time it takes for light to cross through a hydrogen molecule, which is about 247 zeptoseconds, i.e. 247 $\times 10^{-21}$ seconds. Here one of the big sources of uncertainty is what counts as the edge of a hydrogen molecule! But as you can see from the news article, we are deep in the realm of quantum mechanics here, with the randomness associated to that.

Graham Manuell (Aug 04 2024 at 10:45):

@Kevin Carlson It's true that "there’s no nonconstant continuous map from the real line to the space of digits that can display on a stopwatch", but this is not a problem with the real line, but with the stopwatch (or at least with decimal notation). As @John Baez mentions we could just use an analogue clock instead.

Still given the lack of a continuous map there is still the question of how the stopwatch actually works. I looked this up and the crucial component seems to be a comparator, which can be built from a opamp. If $V$ is the voltage obtained from an analogue measurement, then the circuit essentially acts something like $\min(\max(0,10000(V-0.5)),1)$, so if the voltage is a tiny bit above 0.5 then it outputs 1 and if it is a tiny bit below 0.5 it outputs 0. However, there is a very small intermediate region where it has an intermediate value. When designing digital circuits effort is taken to ensure that this intermediate value does not occur in practice, but it is theoretically possible. Moreover, in practice there is noise in the circuit which means values of $V$ near 0.5 actually lead to the circuit rapidly switching between 0 and 1. (This isn't desirable, so real circuits artificially make the output 'sticky'.)

There is no interesting continuous function $[0,1] \to 2^\mathbb{N}$ but there is such a continuous 'nondeterministic' map, which is what the noise ends up giving us.

John Baez (Aug 04 2024 at 11:15):

Thanks for digging a bit more into how a digital stopwatch actually works, @Graham Manuell.

Graham Manuell (Aug 04 2024 at 11:24):

So for a stopwatch specifically maybe it's different because I think it's more about counting oscillations, which are more discrete to start with, if probabilistic/quantum. But what I describe above is how an analogue to digital converter works, which is used in many other cases where we want to translate an analogue measurement into a digital one.

John Baez (Aug 04 2024 at 13:49):

Oscillations are "more digital" because you can count the peaks in a sine wave, but not digital because you can't say for sure exactly where the peaks are, at least in real-world physics, where noise and quantum fluctuations are present.

Nathaniel Virgo (Aug 04 2024 at 13:58):

Nathaniel Virgo said:

I don't know the answer to this question: is that Cantor space invariant to the scale used to measure time? Like if my stopwatch measures time in lunar months instead of seconds, are we measuring the same thing? (I assume the ratio between seconds and lunar months is irrational.)

I think this works as an objection to the idea that time really is a Cantor space as measured by a stopwatch. This might not be what you're asking about at all, but it sounded like it was when you (Kevin) asked "is there any good reason to think that models using the reals rather than the Cantor space is somehow actually "more correct" about how the world is?"

Suppose we both have infinite families of stopwatches that can measure time with greater and greater decimal precision, but your watches measure time in seconds while mine measure time in thirds, where a third is $\sqrt{2}$ seconds. So each of us measures time as a Cantor space, but in different ways.

If we want to say that time really is the Cantor space we measure, then we should expect that the operation of converting between seconds and thirds should be an isomorphism of topological spaces. (At least I think it should. You could object. It depends what "really is" means.) But I don't think it is, because I don't think it's a continuous map.

Kevin Carlson (Aug 04 2024 at 20:41):

OK, I think the point that measurement is a stochastic map from "reality" to outputs as opposed to a continuous one is really helpful.

Kevin Carlson (Aug 04 2024 at 20:43):

I think this is actually a pretty good argument that God plays dice that doesn't strictly rely on knowing about quantum mechanics.

Kevin Carlson (Aug 04 2024 at 20:45):

That is, assuming you believe there's a real manifold sitting out there somewhere in reality, and you very reasonably believe that Nature can't do deterministic discontinuous mappings, then you're pretty much stuck with believing there's irreducible randomness somewhere in the pipeline between the world and your stopwatch.

Kevin Carlson (Aug 04 2024 at 21:06):

Thanks for the idea, Nathaniel, thinking it over.

Graham Manuell (Aug 05 2024 at 06:39):

Kevin Carlson said:

I think this is actually a pretty good argument that God plays dice that doesn't strictly rely on knowing about quantum mechanics.

I really like this kind of argument, but I think it actually depends quite delicately on how the stopwatch works, because we need to know how the stopwatch behaves in these intermediate situations which don't necessarily happen in practice. Like conceivably the stopwatch could interpolate between two digits for a very very short period of time when 0.9 changes to 1.0.

John Baez (Aug 05 2024 at 08:09):

Yes, these are the two choices I see:

I do find it mildly interesting to think about how nature accomplishes a mapping from a continuum of states (like a manifold or a projectivized Hilbert space) to a finite set of "meter readings".

One way is that the finite set of meter readings is really a finite collection of disjoint open sets whose union is not the whole set of meter states: there are also other states where the meter doesn't have a well-defined reading. For example if you are looking at a digital watch, there are short stretches of time when it doesn't show any number at all.

Another way uses randomness. You can have a nonconstant stochastic map from $\mathbb{R}$ (for example) to a finite set. This covers the sort of randomness that appears in classical mechanics. There is also a quantum analogue of this phenomenon.

Chad Nester (Aug 05 2024 at 08:16):

I'm certainly not an expert, and have never heard of some of the flavours of real number mentioned here. That said, my feeling is that calling the real numbers "numbers" is misleading. For me the only "real" numbers are the natural numbers, and the real numbers don't seem to be the same kind of thing. They contain too much!

The moment I think of real numbers as infinite streams, however, any anxiety I have about them disappears. I don't really know why this is.

John Baez (Aug 05 2024 at 08:26):

The ship has sailed centuries ago on calling real numbers 'numbers'. If you say 'number' to a physicist or mathematician, without qualification, about 90% of them will assume you mean a real number or complex number. For professional mathematicians, about 90% of them will have taken a course on advanced calculus or real analysis that explains the usual axiomatization of the real numbers as a complete ordered field, working nonconstructively, and builds up calculus starting from there.

So, while it's fine to have anxiety about these real numbers (they're uncountable, the completeness axioms quantifies over subsets, etc.), and it's fine to advocate substitutes, I don't think it's realistic to hope people stop calling them numbers. (But I guess you weren't actually expecting that to happen.)

Chad Nester (Aug 05 2024 at 08:35):

Yes, I'm not trying to advocate for anything or convince anyone. This is, however, my own personal answer to the question of "what are real numbers?". I feel quite a bit better about them if I think about them as infinite streams, so I'd like that think that that's what they "truly are".

Matteo Capucci (he/him) (Aug 05 2024 at 13:17):

It is objectionable to use 'numbers' for things that famously can't be enumerated :P but I agree with John that that ship has sailed long ago.

Anyway, I kinda agree than treating reals like 'more accurate' natural numbers feels iffy. My preferred way to think about them is geometrically: the reals are, first of all, a geometric entity, the line. I don't think the stream interpretation is very far off a geometric one: the stream is a way to get construct a real number by increasingly zooming in on the line.

Matteo Capucci (he/him) (Aug 05 2024 at 13:20):

Yesterday I was watching this YouTube video and it struck me how there seems to be two notions of number, cognitively speaking: very small and large, the latter linked to spatial reasoning. I wonder if this is where the discrete/continuous divide arises... This is just armchair speculation though.

Peva Blanchard (Aug 05 2024 at 13:30):

the reals are, first of all, a geometric entity, the line

This thread made me think about this exact picture. But something that troubles me is multiplication. Geometrically, multiplying two lines gives a rectangle. I vaguely remember that it was an actual breakthrough in algebra to consider that lines and rectangles (and higher cube-like volumes) can be put on an equal footing. Does someone know more about this ? (from a math history perspective)

John Baez (Aug 05 2024 at 14:11):

I've heard the Greeks understood multiplication geometrically, and implicitly thought of a product of lengths as having units of area, so that something like $2x^2 + 3x + 1$ is dimensionally invalid. James Dolan explained to me how algebraic geometry went back to taking units seriously by re-emphasizing homogeneous polynomials, which pushes us into the arms of projective space. We wrote a little intro to James' ideas on how that plays out in category theory. But I don't know exactly when, sometime in the Renaissance perhaps, mathematicians temporarily freed themselves from the tyranny of dimensional analysis. That would be good to understand!

Matteo Capucci (he/him) (Aug 05 2024 at 15:54):

Al Kwarizmi already summed areas and lengths without problem, so it's at least since the middle ages

Matteo Capucci (he/him) (Aug 05 2024 at 15:55):

You can multiply lengths by pure numbers though, and still get lengths. Indeed Eudoxus' reals are conceived as ratios of lengths. In modern jargon we'd say lengths are an R-torsor.

John Baez (Aug 05 2024 at 15:57):

In modern jargon we'd say nonzero lengths are an $\mathbb{R^\times}$-torsor, where $\mathbb{R}^\times$ is the multiplicative group of nonzero real numbers.

John Baez (Aug 05 2024 at 15:58):

But I'd prefer to say something a bit stronger.

John Baez (Aug 05 2024 at 15:59):

I talked about it my page on torsors.

Finally, one more remark for people who want to go further. Near the beginning of this essay, I said "as soon as we pick units of temperature, temperatures are elements of an R-torsor". We need to pick units of temperature to know what it means to "add 1" to a temperature. So, where should we think of temperatures a living before we pick units? We should think of them as living on a line whose symmetries include not just translations but also dilations - in other words, the "stretchings" or "squashings" that result from a change of units. Picking an origin reduces the symmetry group to just dilations - and indeed, there's a distinguished choice of origin, namely absolutely zero. Picking units reduces the symmetry group to just translations, giving us an R-torsor - and indeed, there's a distinguished choice of units, namely Planck units. Picking both lets us think of temperatures as real numbers. This combination of translations and dilations arises because R is not just a group, but a ring. So, there's a more sophisticated concept than that of "torsor" allowing both translations and dilations whenever you start with a ring.

John Baez (Aug 05 2024 at 16:01):

That was a bit vague, but I was trying to point out that we can define some sort of "ring-torsor" thing that's like a ring where you've forgotten which points are the additive and multiplicative identity elements, just as the usual "group-torsor" is a like a group where you've forgotten the identity element.

John Baez (Aug 05 2024 at 16:02):

I think when we talk about "lengths" we have a distinguished additive identity 0 already: we can all agree on what's zero length. But when we talk about "points on a line", where the line has both translation and dilation symmetries, we're talking about points of a "ring-torsor".

John Baez (Aug 05 2024 at 16:02):

Anyway, I'm probably being too lazy to express myself clearly!

Kevin Carlson (Aug 05 2024 at 17:28):

Peva Blanchard said:

But something that troubles me is multiplication. Geometrically, multiplying two lines gives a rectangle. I vaguely remember that it was an actual breakthrough in algebra to consider that lines and rectangles (and higher cube-like volumes) can be put on an equal footing. Does someone know more about this ? (from a math history perspective)

Well, about the multiplication of two lines being a rectangle, Euclid shows how to make the product of two lines into a line by, I think, giving a rectangle of one side length unity with the same area as a given rectangle. In particular Euclid does have the whole ordered field structure of real numbers transferred over to lines; that didn't have to wait for the early moderns to give up on dimensionality or anything.

Ulrik Buchholtz (Aug 05 2024 at 18:26):

@Peva Blanchard have a look at Bos' Redefining Geometrical Exactness, esp. Ch. 8 (on Viète's “New Algebra”, in which only magnitudes of the same dimension can be added) vs Ch. 15+21 in Part II (on Descartes, who eliminated the homogeneity requirement). The choice of a unit is indeed arbitrary in geometry, and people worried about it at the time (see Fn. 21.14 on Debeaune who in 1649 discussed that the choice of unit influences how $a$ (a side length) compares to $a^2$ (the side length of a rectangle with other side the unit and area $a^2$).

John Baez (Aug 05 2024 at 21:33):

Kevin Carlson said:

Well, about the multiplication of two lines being a rectangle, Euclid shows how to make the product of two lines into a line by, I think, giving a rectangle of one side length unity with the same area as a given rectangle

That sounds like the good way to do it. I'd like to see what he said by that, and also see what the Greeks said about solving quadratic equations. Since they didn't have variables like $x$, and they didn't have negative numbers, and it seems they mainly reasoned geometrically, I guess it would have to look quite different from what we say. Did they take each of the terms we'd call $ax^2, bx$ and $c$ and interpret them as the area of a rectangle?

John Baez (Aug 05 2024 at 21:35):

I'm also really interested in what Viète and Debeaune and Descartes said about these issues - they would have read Euclide - but I'm locked out of Springer's paywall right now.

Kevin Carlson (Aug 05 2024 at 22:00):

I don't know about the French early modernists, but looking back into it, Euclid doesn't actually prove quite what I said, because the Greeks weren't into fixing a unit length. Instead, in Book VI Proposition 12 he constructs, given lengths $A,B,C$, a fourth length $D$ such that $C:D=A:B,$ using similar triangles. It's a quick leap for Descartes or somebody to then set $A=1$ to get $D=BC$!

John Baez (Aug 05 2024 at 22:07):

Thanks! The business of not setting $D = 1$ is important in modern algebraic geometry, where homogeneous equations like $C/D = A/B$ define projective varieties.

John Baez (Aug 05 2024 at 22:08):

Later you can set $D = 1$ and get an affine variety, but this leaves out the points where $D = 0$.

Ryan Wisnesky (Aug 06 2024 at 04:54):

There's a long history of 'analog computation' within CS. Ditto for "real computation": https://en.wikipedia.org/wiki/Real_computation , and what these computer could in principle compute is tied to unsolved physics problems. Some of the ideas, such as asynchronous chip design, are even back in vogue as ways to save power. And folks are still trying to beat IEEE floating point, eg. http://www.johngustafson.net/pdfs/BeatingFloatingPoint.pdf

Robin Piedeleu (Aug 06 2024 at 10:04):

The discussion above on stopwatches implementing some stochastic mappings from the reals to some finite set of states reminded me of this quanta article (and the paper cited within) on the thermodynamics of clocks. Though my understanding of this work is limited, one key takeaway appears to be that

The greater a clock’s accuracy, the more energy it dissipates and the more entropy it produces in the course of ticking.

So reducing the randomness of the time interval between consecutive ticks costs more and more energy and produces more and more entropy; in the limit, perfectly accurate clocks are thus not physically realisable.

If I understand correctly, this also suggests that we can think of a clock as a stochastic process with values in, e.g., some two element set $\{0,1\}$, that is, as a stochastic map $C$ from the reals to $\{0,1\}$, such that the time between any pair of consecutive ticks (changes in the value of $C$) "is statistically independent from, and identically distributed to, the time between any other pair of consecutive ticks", and has finite standard deviation (i.e., fixed accuracy).

John Baez (Aug 06 2024 at 10:59):

That would be a particular well-behaved kind of clock - idealized in that we'll never get perfect i.i.d. random variables, but much more realistic than a 'perfect' clock.

I haven't read the argument in Autonomous quantum clocks: does thermodynamics limit our ability to measure time? that aims to show

The greater a clock’s accuracy, the more energy it dissipates and the more entropy it produces in the course of ticking.

I like such arguments when they make sense to me, and I get frustrated when they don't. I'll give it a try.

People who enjoy quantum mechanics may enjoy my article

• The time-energy uncertainty relation

which discusses limitations on a possible "time operator" in quantum mechanics. Ideally this would be a time-dependent observable $T(t)$ that evolves in time via

$\displaystyle{ \frac{dT(t)}{dt} = 1 .}$

In practice we usually need to settle for less.

Robin Piedeleu (Aug 06 2024 at 11:08):

The authors seem to justify this idealisation as follows (on p.4 of the arXiv version of the paper):

For our model of the autonomous clock, we assume that after each spontaneous emission event, the entire pointer is reset to its initial state — specifically, a product state with the ladder in its ground state and the engine qubits in equilibrium with their respective baths. This approximation is valid in the weak-coupling limit, where the engine qubits are minimally perturbed by their interaction with the ladder. The ticks of the clock can therefore be described as a renewal process, i.e. the time between any pair of consecutive ticks is statistically independent from, and identically distributed to, the time between any other pair of consecutive ticks.

John Baez (Aug 06 2024 at 11:18):

I'm sad to see that the paper is treating a specific model of a clock rather than proving a general result. This is still a good thing, but since I'm a mathematician I prefer general results (like the time-energy uncertainty relation for example). Sometimes studying specific models is a good way to start. For example, someone may try to "get around" the result in this paper and discover they either can or can't, and that can lead to more papers, and ultimately a general result. But I'll wait until some more general results come along before I put thought into this.

A machine can't really "reset" a quantum bit without changing something else too - I wonder if the authors are taking this into account, or idealizing this fact away.

John Baez (Aug 07 2024 at 08:03):

Peva Blanchard said:

Geometrically, multiplying two lines gives a rectangle. I vaguely remember that it was an actual breakthrough in algebra to consider that lines and rectangles (and higher cube-like volumes) can be put on an equal footing. Does someone know more about this ? (from a math history perspective)

By coincidence I bumped into this:

• Keith Devlin, Algebra: it's powerful, but it's not what it once was.

For most of history, what we today think of as a number (i.e., an object, or noun) was in fact the adjective in a multitude-species pair. People conceived numbers as real-world objects viewed from the perspective of quantity; so numbers came attached to objects (though for mathematicians they could be abstract objects). We are familiar with this today with currency and geometric angles. We have monetary amounts such as “5 dollars and 20 cents” or angles such as “15 degrees and 30 seconds”. The numerals in those expressions are adjectives that modify the nouns (those nouns being dollars, cents, degrees, and seconds, respectively).

Those multitude-species pairs were the entities that equations were made up of; they were their “numbers”. For example, the following pre-modern quadratic equation comes from an eleventh-century Arabic algebra textbook

“eighty-one shays equal eight māls and one hundred ninety-six dirhams”.

Here the English version consists of translations of the Arabic words from that textbook, apart from the terms “māls”, “dirhams”, and “shays”. The word shay referred to the unknown in the equation, māl is the term they used to denote the square of the unknown, and dirham denoted the unit term. The plural “s”s in our presentation of the equation are English additions (hence not italicized); Arabic does not designate plurals that way.

John Baez (Aug 07 2024 at 08:06):

Later in the article:

The key to today’s algebra was the creation of an abstract number system, specified by axioms, a process that was not completed until the early twentieth century. A valuable first step towards that solution was made by several seventeenth century mathematicians who defined numbers-as-objects from the multitude-species pairs that had served for so long.

Newton, for example, wrote down the proposal below in one of his Notebooks (1670). The species he was working with for calculus were line segments, which he called Quanta. To develop calculus, he looked at line segments from the perspective of their length; those were his multitude-species “numbers”, which he wanted to break free of. He wrote:

Number is the mode or affection of one quantum compared to another which is considered as One, whereby its Ratio or Proportion to that One is expressed. [Thus b/a  is the number expressing the ratio of b to a.]

The brackets are his. He wrote “b/a” is vertical-stack form, not inline as here. His key claim was: “b/a is the number.” He had defined a ratio (hence a dimensionless object) to be his number concept.

Defining numbers to be ratios of Quanta was equivalent to adopting a unit Quantum. He formulated other variants in other Notebooks.

Ulrik Buchholtz (Aug 07 2024 at 08:13):

John Baez said:

I'm also really interested in what Viète and Debeaune and Descartes said about these issues - they would have read Euclide - but I'm locked out of Springer's paywall right now.

I can send you a copy of Bos' book, if you haven't gotten it yet, but it doesn't have many direct quotations. Descartes' Geometry is available with a facing English translation at the Internet Archive. Book I starts with the relationship between arithmetic and geometry. For instance, on p. 5 he writes:

Here it must be observed that by $a^2$, $b^3$, and similar expressions, I ordinarily mean only simple lines, which, however, I name squares, cubes, etc., so that I may make use of the terms employed in algebra.

It should also be noted that all parts of a single line should always be expressed by the same number of dimensions, provided unity is not determined by the conditions of the problem. Thus, $a^3$ contains as many dimensions as $ab^2$ or $b^3$, these being the component parts of the line which I have called $\sqrt[3]{a^3-b^3+ab^2}$. It is not, however, the same thing when unity is determined, because unity can always be understood, even where there are too many or too few dimensions ; thus, if it be required to extract the cube root of $a^2b^2-b$ we must consider the quantity $a^2b^2$ divided once by unity, and the quantity $b$ multiplied twice by unity.

John Baez (Aug 07 2024 at 08:20):

Thanks very much - this looks better than Bos' book for what I want. It's great to see Descartes explicitly grappling with this issue. By the time we get to 20th-century math classes, variables are unitless and nobody even bothers to mention this. Then, when students take physics classes, they can be surprised by how physicists use dimensionful variables and an equation like

$T = 275$

is illegitimate if $T$ is temperature. Similarly for chemistry and other sciences.

Kevin Carlson (Aug 07 2024 at 20:33):

This also reminds me of the discussion in Mayberry's The Foundations of Mathematics in the Theory of Sets of the Greek attitude toward number, which he argues is almost indistinguishable from our concept of a finite set. (A number of sheep, for instance.)

John Baez (Aug 08 2024 at 08:51):

So decategorification hadn't fully set in. Interesting!

One difference is that in sufficiently early Greek mathematics, "one" was not considered a number - indeed it was considered the opposite of number (multitude). Also there seems to have been a special attitude about "two", perhaps because Greek had singular, dual and plural. For example they said 2 was not prime because it's even, and primes (obviously) are not even.

John Baez (Aug 08 2024 at 08:53):

So my theory is that in Indo-European cultures, first they thought numbers were 3, 4, 5, .... Then they decided 2 was also a number. Then they decided 1 was also a number. Then they decided 0 was also a number. Then they decided -1 was a number, and the floodgates opened.

Madeleine Birchfield (Aug 08 2024 at 17:06):

Were the positive rational numbers considered numbers by the Greeks, or were they considered something else at first? If the latter, when did positive rational numbers become numbers?

John Baez (Aug 08 2024 at 21:18):

That's a great question. I don't know the answer. When I've been saying "numbers" in connection with ancient Greek mathematics, the word I mean is arithmoi. They may have considered rational numbers to be ratios of arithmoi. But I don't know. It's also good to remember that what people often call ancient Greek mathematics goes from at least Pythagoras (born 570 BC) to Ptolemy (died 170 AD); this covers a big span of times and places.

John Baez (Aug 09 2024 at 10:37):

@Madeleine Birchfield - it seems that among the ancient Greeks, the issue of natural numbers versus rationals gets intertwined with dimensional analysis: see the Mathstodon thread starting here.

Apparently following Eudoxus Euclid defined 'quantities' to be ratios of 'magnitudes' with like dimension, where magnitudes are dimensionful but quantities are dimensionless. The'quantities' thus obtained seem to all be positive real numbers in the smallest subfield of $\mathbb{R}$ closed under solving quadratic equations with real roots, since straightedge-and-compass constructions can solve quadratic equations but not most higher-degree polynomial equations.

John Baez (Aug 09 2024 at 10:41):

I momentarily forgot the jargon: these are called the nonnegative constructible numbers.

John Baez (Aug 14 2024 at 15:43):

The psychiatrist's approach: