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October 23, 2006

Knowledge of the Reasoned Fact

Posted by David Corfield

In a comment I raised the question of what to make of our expectation that behind different manifestations of an entity there is one base account, of which these manifestations are consequences.

If I point out to you three manifestations of the normal distribution - central limit theorem; maximum entropy distribution with fixed first two moments; approached by distribution which is the projection onto 1 dimension of a uniform distribution over the nn-sphere of radius n\sqrt{n} as nn increases - it’s hard not to imagine that there’s a unified story behind the scenes.

Perhaps it would encourage a discussion if I put things in a contextual setting.

On page 51 of Mathematical Kinds, or Being Kind to Mathematics, I mention the idea of Aronson, Harré and Way (Realism Rescued: How Scientific Progress is Possible, Duckworth, 1994), that one of the crucial functions of sciences is to organise entities into a hierarchy of kinds. One important idea here, found in other writings of Rom Harré, is that the use of first-order logic by logical positivists and then logical empiricists to analyse scientific reasoning has been a disaster. What these authors realised was that something important had been lost from earlier Aristotelian conceptions.

In the Posterior Analytics, Aristotle claims that are four kinds of thing we want to find out:

  • Whether a thing has a property
  • Why a thing has a property
  • Whether something exists
  • What kind of thing it is

As you can see, these four types come in two pairs, the second of the pair asking a deeper question than the first. Indeed, the second and fourth questions ask about the cause of something and its properties, cause being taken in Aristotle’s broad way. In fact, this is broad enough that he has no qualm in discussing examples from astronomy (‘Why do planets not twinkle? Because they are near.’), mathematics (‘Why is the angle in a semicircle a right-angle?’) and everyday life (‘Why does one take a walk after supper? For the sake of one’s health. Why does a house exist? For the preservation of one’s goods.’). This talk of cause in mathematics continued for many centuries. For example, as we learn from Mancosu’s book, Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (OUP, 1996), mathematicians considered that giving a proof of a result by a reductio argument was not to give its ‘cause’.

The salient distinction here is that of Aristotle between ‘knowledge of the fact’ and ‘knowledge of the reasoned fact’. Aristotle gives this example to illustrate the difference:

Planets are near celestial bodies.
Near celestial bodies don’t twinkle.
Therefore, planets don’t twinkle.

Planets don’t twinkle.
Celestial bodies which don’t twinkle are near.
Therefore, planets are near.

To give an easy example of Aristotle’s distinction in mathematics:

nn is an even number.
Even numbers expressed in base 10 end in 0,2,4,6 or 8.
nn ends in 0,2,4,6 or 8.

nn expressed in base 10 ends in 0,2,4,6 or 8.
Numbers expressed in base 10 which end in 0,2,4,6 or 8 are even.
nn is even.

It’s not that the second syllogism is wrong, but rather that it hasn’t got the ‘causal’ ordering correct. Explanation is about tapping into the proper hierarchical organisation of entities. Your ability to do this is what needs to be tested, as here in this account of an MIT mathematics oral examination (in particular, the Symplectic Topology section).

The idea that mathematics has a causal/conceptual ordering seems to be yet more radically lost to us than the counterpart idea in philosophy of science. It’s at stake in the example I give in my paper Dynamics of Mathematical Reason (pp. 11-13), where singular cohomology goes from its being a cohomology theory which happens to satisfy the excision and other axioms, to its being a cohomology theory because it satisfies the excision and other axioms.

Now, should we expect convergence to a single ordering?

Posted at October 23, 2006 1:25 PM UTC

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Re: Knowledge of the Reasoned Fact

[M]athematicians [in the 17th century] considered that giving a proof of a result by a reductio argument was not to give its ‘cause’.

So only constructive proofs give causes?

This makes sense if we recall the computational content of constructive proofs; they can be turned systematically into computer programs. For example, a constructive proof of the infinitude of primes automatically (at least if you write it out formally in type theory) gives an algorithm that, given a finite list of natural numbers (including being given its length), calculates a prime number that is not on the list. More interestingly, a constructive proof of the uncountability of the continuum, applied to a (computable) enumeration of the algebraic numbers, automatically computes a (computable) transcendental number. So the “reason” that transcendental numbers exist is that we have a way to construct them.

However, I don’t believe that this can be all that there is to it. Consider a reductio proof of Lagrange’s Theorem that every natural number may be decomposed as the sum of four squares. This proof can easily be made constructive by checking that there are only finitely many possibilities (given any specific number) for such a decomposition (since the summands can never be larger than the desired sum). However, the algorithm that results from this step is terribly inefficient; it simply searches through all possible summands until a solution is found! Surely the “cause” (whatever that means) of this theorem must be something more than that, if you search through the possibilities, you eventually succeed. You still want to ask why you will succeed, and the reductio, even constructivised, gives no help.

Much more satisfying is a proof (the usual one is already constructive, not relying on reductio) using unique factorisation in the quaternions. (How close is this to Lagrange’s original proof? I don’t know.) I’m not sure that this really deserves to be called “cause”, but it strikes me as much more of an “explanation” at least!

(Personally, I don’t believe in causation. I’ve never been convinced by any philosopher, from Aristoteles to David Lewis, that the subject belongs in metaphysics (as opposed to epistemology) at all. I believe in a universe described by relativity theory with a low-entropy big bang, but the consequences of this fall far short of most people’s understanding of cause and effect; it certainly doesn’t apply to the theorem above!)

Posted by: Toby Bartels on October 23, 2006 7:58 PM | Permalink | Reply to this

Re: Knowledge of the Reasoned Fact

I’ve never been convinced by any philosopher, from Aristoteles to David Lewis, that the subject [causality] belongs in metaphysics (as opposed to epistemology) at all.

Sounds like your view is close to my friend Jon Williamson, who holds a view he terms ‘epistemic causality’ (see some papers here): “Causality, then, is a feature of our epistemic representation of the world, rather than of the world itself.”

Remember that what is translated as ‘cause’ in Aristotle had a different range of connotations from today’s usage. This must be so for him to have described physical and mathematical ‘causation’ as though they were cases of the same thing.

Posted by: David Corfield on October 23, 2006 8:35 PM | Permalink | Reply to this

Re: Knowledge of the Reasoned Fact

I find the idea of epistemic causality convincing, because causality seems to me like a kind of convenient shorthand, or maybe a pedagogical device. When we talk about physical causality, like “The glass fell off the table because I pushed it”, we’re already bringing in plenty of epistemic constructs. Adding the judgement that one condition, described in terms of those constructs, is the cause of a later one, seems again to be an epistemic act.

A lower-level (but still theory-laden) description of the same scenario might say something of the form “the reason the configuration of matter-fields in region X of spacetime is such-and-such… is that the configuration on a Cauchy surface in the past light-cone of X was so-and-so”. If the glass takes about a second to fall, that Cauchy surface in the past will be huge, certainly including the whole of the Earth - but saying that everything in the whole world is the cause of any act taking more than a second is not good pedagogy, or helpful for normal thinking. Moreover (assuming physical determinism - more theory), it would be an equally good explanation to look at the future light-cone, so this isn’t much like a causal description.

Going from this low-level physics description to the much clearer causal story about the glass being pushed requires “registering” certain parts of the field as objects (I’m getting this language from Brian Cantwell Smith), and then identifying which features of the past history are the salient ones in describing what those objects do. And both registration and saliency strike me as having a very epistemic flavour.

It seems like every causal story similarly involves throwing out a lot of distracting information and focusing on salient facts, even though one can almost always imagine some (low-probability) scenario where the salient “causes” are the same, but the effects are different (e.g. a bird decides to fly through the room, and just happens to knock the glass back up onto the table), so any causal story would only be a rough sketch, with the implied proviso, “and everything else can be ignored”.

For mathematical objects, where we don’t have time dependence, I guess I’d have to take the same position. A number n being even is connected to all sorts of facts about n, but only some of them feel clear as “causes” of that fact. Just like before, that feels to me like a decision, rather than some kind of metaphysical given.

To make this analogy between a concrete occurrence and a logical deduction, I suppose I’m taking it for granted that concepts like “number” and “even” are epistemic structures we build to organize real-world experiences, just like “glass” and “table” (and “matter-field” and “Cauchy surface” for that matter). So then saying “n is even” is really making a blanket statement about a bunch of, still concrete, features of the world. Lumping those together as similar through a process of abstraction seems again to be an epistemic act. I’m open to the idea that numbers and evenness (or indeed tables) are actually Platonic forms, or some such idealist position - but I’ve never been persuaded of it yet.

Posted by: Jeff Morton on October 24, 2006 3:22 AM | Permalink | Reply to this

Re: Knowledge of the Reasoned Fact

A number n being even is connected to all sorts of facts about n, but only some of them feel clear as “causes” of that fact.

I think this puts into words some of my uneasiness about that section of the original post.

I think it gets a bit easier from the structuralist viewpoint, where the natural number system is just the structure encoded by the Peano axioms. Divisibility properties (like evenness) are an extra layer of structure on top of these axioms: we can use any model of the natural number structure to construct a model of the ring structure. Decimal expansions are a further – and a more arbitrary – layer of structure. Causality seems to flow from divisibility properties to divisibility tests because the former structure is more primitive than the latter.

Of course, this assumes that my viewpoint on which structure is more primitive is an objective one.

Posted by: John Armstrong on October 24, 2006 4:20 AM | Permalink | Reply to this

Re: Knowledge of the Reasoned Fact

Dear David,

It appears to me that for some aspects of the issues you raise it would make sense to consider

a) Very basic mathematical facts that we learn in school or early college

b) Naive understanding (by children/students) of mathematical phenomena, and of causality and reasons for them.

(Perhaps somewhat like Chomskian linguistics.)

e.g. (school stuff, ask children, compare to what professionals say) what is the reason for

1. 5+8 = 8+5

2. 6*5 =5*6

(Is 1. part of the reason for 2.?? is there a common reason??)

3. 42 - 19 = 23.

High school:

4. why a^b is not equal b^a ?

5. The sum of angles in a triangle 180 degrees.

College:

6. square root of 2 is not a rational number

7. A continuous function that takes a positive value at a and a negative at b takes 0 in between.

And a question that bothered me as a student and I never heard any satisfying answer to since:

8. Why is it that a real function that has a derivarive at any point may fail to have second derivative but a complex function once having a derivative at every point has second (and third etc) derivatives??

Anyway, this looks like a good place to compare insights of laymen and “professionals” and to consider concretely some examples like suggested above.

Posted by: Gina on October 24, 2006 4:19 AM | Permalink | Reply to this

Re: Knowledge of the Reasoned Fact

That’s a great idea, but for the reality of the situation. I don’t know the XML tag for pessimism, so I’ll just take it as understood.

1. 5+8 = 8+5

2. 6*5 =5*6

(Is 1. part of the reason for 2.?? is there a common reason??)

In much of the country, the “why” is “because that’s what the times tables say. It’s all arithmysticism.

3. 42 - 19 = 23.

Kids may be able to muster up, “because 19+23 = 42”.

4. why a^b is not equal b^a ?

5. The sum of angles in a triangle 180 degrees.

I think far too few high school students know these, let alone have any explanation. In the first case, I can attest that college students (Ivy leaguers, no less) don’t know it. In the second, it’s one of those things most people can quote but few can justify beyond an appeal to authority in the form of a high school geometry teacher.

6. square root of 2 is not a rational number

Those who understand what a rational number is can probably say something halfway sensible. Those few.

7. A continuous function that takes a positive value at a and a negative at b takes 0 in between.

Here’s the first one that I think even has a naïve understanding.

8. Why is it that a real function that has a derivarive at any point may fail to have second derivative but a complex function once having a derivative at every point has second (and third etc) derivatives??

And here I think all hope is lost. Nobody gets to the point of even learning the definition of a complex derivative without leaving the intuitive/naïve realm.

Chomskian linguistics works because everyone uses language and it seems perfectly natural to ask “why” questions about it. By and large most people are content to think of mathematics at even the most basic level as some sort of esoterica. The only reason any of it is true for most people is that that’s what they were taught, if they remember it at all.

Posted by: John Armstrong on October 24, 2006 5:33 AM | Permalink | Reply to this

Re: Knowledge of the Reasoned Fact

Gina asked about the following:

1. 5+8 = 8+5

I think I have an intuitive justification for this, in terms of piles of pebbles, which one could reasonably call “naïve”. For this,

4. why a^b is not equal b^a ?

I’m not so sure. It’s easy to give an example where a bb aa^b \neq b^a which a schoolchild could check, but I don’t have much confidence in my ability to explain why such examples must exist.

Whether that matters, of course, depends upon the intellectual curiosity of the schoolchild. We’re in luck if she asks me a question which stumps me entirely!

5. The sum of angles in a triangle 180 degrees.

Thanks to Project Mathematics!, I always imagine extending the triangle’s sides to make the vertical angles, and then shrinking the entire triangle down to a point…

I side with John Armstrong on points six and eight. As to this one,

7. A continuous function that takes a positive value at a and a negative at b takes 0 in between.

I think we have segued into the realm of assertions which are easy to support with a pencil sketch but explode into subtleties as soon as you try to introduce any notion of rigor. How can we explain what a “continuous function” is? A textbook might do so as follows:

In everyday speech, a ‘continuous’ process is one that proceeds without gaps of interruptions or sudden changes. Roughly speaking, a function y=f(x)y = f(x) is continuous if it displays similar behavior, that is, if a small change in xx produces a small change in the corresponding value f(x)f(x).

In fact, a textbook does do so, specifically G. F. Simmons’s Calculus with Analytic Geometry (McGraw-Hill, 1985). This statement is “rather loose and intuitive, and intended more to explain than to define.” To give a real definition, we break out the machinery of limits and begin employing deltas and epsilons, as Tom Lehrer describes here:

  • Tom Lehrer, “There’s a Delta for Every Epsilon”, performed for Irving Kaplansky’s 80th birthday celebration (19 March 1997).

However, as Keith Devlin has written,

With limits defined in this way, the resulting definition of a continuous function is known as the Cauchy–Weierstrass definition, after the two nineteenth century mathematicians who developed it. The definition forms the bedrock of modern real analysis and any standard “rigorous” treatment of calculus. As a result, it is the gateway through which all students must pass in order to enter those domains. But how many of us manage to pass through that gateway without considerable effort? Certainly, I did not, and neither has any of my students in twenty-five years of university mathematics teaching. Why is there so much difficulty in understanding this definition? Admittedly the logical structure of the definition is somewhat intricate. But it’s not that complicated. Most of us can handle a complicated definition provided we understand what that definition is trying to say. Thus, it seems likely that something else is going on to cause so much difficulty, something to do with what the definition means. But what, exactly?

Devlin advances the idea that going from the intuitive statement — “a line you can draw without picking up your pencil” — to the Cauchy–Weierstrass definition is not just a matter of refinement or increasing the “rigor”, but instead a fundamental change from a dynamic to a static view:

Let’s start with the intuitive idea of continuity that we started out with, the idea of a function that has no gaps, interruptions, or sudden changes. This is essentially the conception Newton and Leibniz worked with. So too did Euler, who wrote of “a curve described by freely leading the hand.” Notice that this conception of continuity is fundamentally dynamic. Either we think of the curve as being drawn in a continuous (sic) fashion, or else we view the curve as already drawn and imagine what it is like to travel along it. […] When we formulate the final Cauchy–Weierstrass definition, however, by making precise the notion of a limit, we abandon the dynamic view, based on the idea of a gapless real continuum, and replace it by an entirely static conception that speaks about the existence of real numbers having certain properties. The conception of a line that underlies this definition is that a line is a set of points. The points are now the fundamental objects, not the line. This, of course, is a highly abstract conception of a line that was only introduced in the late nineteenth century, and then only in response to difficulties encountered dealing with some pathological examples of functions.

When you think about it, that’s quite a major shift in conceptual model, from the highly natural and intuitive idea of motion (in time) along a continuum to a contrived statement about the existence of numbers, based on the highly artificial view of a line as being a set of points. When we (i.e., mathematics instructors) introduce our students to the “formal” definition of continuity, we are not, as we claim, making a loose, intuitive notion more formal and rigorous. Rather, we are changing the conception of continuity in almost every respect. No wonder our students don’t see how the formal definition captures their intuitions. It doesn’t. It attempts to replace their intuitive picture with something quite different.

All of these passages have been quoted from the following:

In other words, the example which seems easiest to demonstrate intuitively and pictorially leads you to real issues when you try to formalize it.

Posted by: Blake Stacey on October 24, 2006 3:10 PM | Permalink | Reply to this

Re: Knowledge of the Reasoned Fact

One of the subtleties mentioned is that the Cauchy-Weierstrass definition of continuity also implicitly takes on the job of capturing the completeness property of the real numbers, which the original intuition doesn’t try to do. One can imagine (as many did, pre-Pythagoras) that all time quantities are rational numbers. In which case, if a continuous curve is one that can be drawn through time without any jumps, then statement 7 is false.

Posted by: Jeff Morton on October 25, 2006 12:21 AM | Permalink | Reply to this

Continuity.

[T]he Cauchy-Weierstrass definition of continuity also implicitly takes on the job of capturing the completeness property of the real numbers […].

Devlin’s article mentions this towards the end:

[The] epsilon-delta statement […] does not eliminate (all) the vagueness inherent in the intuitive notion of continuity. Indeed, it doesn’t address continuity at all. Rather, it simply formalizes the notion of “correspondingly” in the relation “correspondingly close.” In fact, the Cauchy-Weierstrass definition only manages to provide a definition of continuity of a function by assuming continuity of the real line!

(I’m not sure if you meant to allude to this, Jeff, so I’m making it explicit anyway.)

Jeff again:

One can imagine […] that all time quantities are rational numbers. In which case, if a continuous curve is one that can be drawn through time without any jumps, then statement 7 is false.

For example (while I’m making things explicit), let f(x) be x2 − 2.

More fully, Jeff wrote:

One can imagine (as many did, pre-Pythagoras) that all time quantities are rational numbers.

But of course, it was not until Pythagoras that anybody knew that my function f does not cross the real line! (Still, one can reasonably imagine thus even after Pythagoras.)

Actually, it seems pretty ironic (to me) that #7, while seeming to many the most intuitive, is also the most doubtful! Besides frankly different mathematical interpretations of the intuition of continuity (like using rational numbers instead of real numbers), it’s also possible to understand the same mathematical statement differently.

Since I do constructive mathematics sometimes, #7 jumped out to me as (potentially) wrong. To be precise, it is not provable in a neutral constructive setting (like Errett Bishop’s constructive analysis), and it is flatly refutable in more restricted constructive settings (including both Brouwer’s intuitionistic analysis and the Russian school of constructive recursive analysis, for different reasons).

This is interesting (to me, in part) because of all the discussion of static vs dynamic intuitions. The BHK interpretation of constructive logic gives the uniform continuity of a function (which is a bit simpler than pointwise continuity) a dynamic flavour, albeit not the original dynamic intuition of tracing out a path.

To be precise, it invites us to view the ∀ε∃δ statement as the description of a process of transformation (or at least, the claim that one exists) that, given a natural number n (an upper bound on 1/ε), returns a natural number m (giving δ as 1/m). That is, once we decide how closely we want to approximate the value of the function, we apply this transformation to determine how closely we must approximate the argument. The dynamic nature here is not between points on the curve itself but rather between our uses of the function for measurement and calculation.

Posted by: Toby Bartels on October 27, 2006 4:08 AM | Permalink | Reply to this

Re: Knowledge of the Reasoned Fact

Dear all,
I agree with John that a Chomskian study of children’s learning, reasonings and insights about counting, arithmetics and mathematics will have limmited scope compared to the similar study for languages. It may still be useful. Beside the philosophical issues of mathematical causality it can be relevant to the understanding of dyscalculia- learning disabilities related to mathematics (Math Dyslexia).

I would conjecture that children usually understand why sum and product are commutative and the meaning of substractions but the algorithms for arithmetic operations on 2-digit numbers obscure this understanding. (This was the point behind 42-19.)

Moving to naive or intuitive reasoning/causality for higher mathematics, the many ways to understand countinuous functions are fascinating. I would be happy to hear if there is a “reason” or “intuition” behind the miraculous difference between real functions which have a derivative at every point and complex functions with the same property.

Posted by: Gina on October 25, 2006 8:35 PM | Permalink | Reply to this

Re: Knowledge of the Reasoned Fact

Gina writes:

Why is it that a real function that has a derivarive at any point may fail to have second derivative but a complex function once having a derivative at every point has second (and third etc) derivatives?? I would be happy to hear if there is a “reason” or “intuition” behind the miraculous difference between real functions which have a derivative at every point and complex functions with the same property.

There certainly is a reason, and when I teach complex analysis I try to explain it.

After all, this is one of the biggest pleasant surprises in mathematics. In real analysis you have to pay extra for each derivative: for example, most functions that are 37 times differentiable do not have a 38th derivative. But in complex analysis being differentiable once ensures a function is infinitely differentiable! It’s as if you bought a doughnut at a local diner and they promised you free meals for the rest of your life! And some people say there’s no such thing as a free lunch….

So, we have to understand this seeming miracle.

The first fact I try to impress on my students is that for a differentiable function ff on the complex plane, the amount f(x+iy)f(x + i y) changes when you change yy a teeny bit is ii times the amount f(x+iy)f(x + i y) changes when you change xx a teeny bit. Why? Because a tiny step north is ii times a tiny step east, and the derivative is a linear approximation to ff.

That’s simple enough. But, the fact that a step north is ii times a step east makes linearity far more powerful on the complex plane than on the real line, where you could only take, say, 2 steps east, or -3 steps east. Now, just to be differentiable, a function must satisfy a differential equation:

fy=ifx {\partial f \over \partial y} = i {\partial f \over \partial x}

the Cauchy-Riemann equation.

This makes all sorts of great things happen. For one thing, it means we can’t change ff one place without changing it lots of other places: if we mess with it in a tiny neighborhood, it won’t satisfy the Cauchy-Riemann equation at the edge of that neighborhood anymore.

So, differentiable functions on the complex plane are not “floppy” the way differentiable functions on the real line are. You can’t wiggle them around over here without having an effect over there.

In fact, if you know one of these functions around the edge of a disk, you can solve the Cauchy-Riemann equation to figure out what it equals in the middle! So, such a function is like a drum-head: if you take your fingers and press the drum-head down near the rim, the whole membrane is affected.

Indeed, the height of a taut drum-head satisfies the Laplace equation, which also holds for any function satisfying the Cauchy-Riemann equation:

( 2x 2+ 2y 2)f=(x+iy)(xiy)f=0({\partial^2 \over \partial x^2} + {\partial^2 \over \partial y^2}) f = ({\partial \over \partial x} + i {\partial \over \partial y})({\partial \over \partial x} - i {\partial \over \partial y})f = 0

So, the analogy is not a loose one: you really can understand what complex-analytic functions look like - well, at least their real and imaginary parts - by looking at elastic membranes.

And if you do this, one thing you’ll note is that such membranes are really, really smooth. One way to think of it is that they’re minimizing energy, so any “unncessary wiggliness” is forbidden. We can make this precise by noting that the Laplace equation follows from a principle of minimum energy, where energy is

|f| 2dxdy \int\int |\nabla f|^2 \; dx dy

So, the reason why a once differentiable complex function is automatically smooth is that:

1) north is ii times east

so

2) to be differentiable, a function on the complex plane must satisfy a differential equation

and

3) this equation makes the function act like an elastic membrane.

This is a remarkable combination of insights, none particularly complicated, but fitting together in a wonderful way.

Posted by: John Baez on October 26, 2006 1:58 AM | Permalink | Reply to this

Re: Knowledge of the Reasoned Fact

Many thanks, John for this beautiful reason. It looks very appealing and quite different from the way I remembered it. Now, I wonder if your explanation (which is very inspiring) will qualify for being a “causation” (of the kind David asked about) even on a heuristic level. To examine it, we should ask: Does such a reason apply in other cases? Namely, is there any (non trivial) example, or even perhaps a whole large class of examples, where your point 3 applies: a differential equation that forces every differentiable solution to have derivatives of any order.

Posted by: Gina on October 26, 2006 1:00 PM | Permalink | Reply to this

Re: Knowledge of the Reasoned Fact

How perturbable are the Cauchy-Riemann equations? What other local conditions on partial derivatives force global solutions in interesting ways?

Is there something special at play because things are easily expressible in terms of the complex field? Does quaternionic analysis force itself upon you in a similar way?


Posted by: David Corfield on October 28, 2006 8:57 PM | Permalink | Reply to this

Re: Knowledge of the Reasoned Fact

David wrote:

How perturbable are the Cauchy-Riemann equations? What other local conditions on partial derivatives force global solutions in interesting ways?

A lot of basic stuff about existence and smoothness of solutions generalizes from the Cauchy-Riemann equation and Laplaceequation to any “elliptic” PDE, as sketched here. This is one reason Atiyah and Singer were able to generalize the Riemann-Roch theorem from the Cauchy-Riemann operator to all elliptic operators.

But…

Is there something special at play because things are easily expressible in terms of the complex field? Does quaternionic analysis force itself upon you in a similar way?

… there certainly are special features of the Cauchy-Riemann equations, coming from its intimate connection to the complex numbers!

By comparison, quaternionic analysis has been a bust. Several obvious ways of generalizing the concept of analytic function from the complex to quaternionic case give really pathetic results. The good way is due to Fueter. It works not just for quaternions but also Clifford algebras. However, much to my shame, I’ve never really spent the time needed to learn it! And, few other people seem to know it. I can’t tell if it’s unjustly neglected, or really not very interesting.

The last link above leads to a description and excerpt of Tony Sudbery’s paper “Quaternionic Analysis” - by far the best thing to read on this subject. It also has a link to his paper… which, alas, no longer works!

And now I’ve lost my copy.

Posted by: John Baez on October 29, 2006 8:29 AM | Permalink | Reply to this

Re: Knowledge of the Reasoned Fact

Gina wrote:

Many thanks, John for this beautiful reason. It looks very appealing and quite different from the way I remembered it. Now, I wonder if your explanation (which is very inspiring) will qualify for being a “causation” […] Namely, is there any (non trivial) example, or even perhaps a whole large class of examples, where your point 3 applies: a differential equation that forces every differentiable solution to have derivatives of any order?

I’m glad you enjoyed my little explanation. It’s sad that most classes on complex analysis don’t explain this stuff.

Yes, there is a vast class of partial differential equations (PDE) such that any solution automatically has derivatives of arbitrarily higher order! These are the so-called elliptic differential equations. If you talk to experts on PDE, you’ll find they often prefer to concentrate on one of these three kinds:

  • Elliptic: here the classic example is the Laplace equation 2fx 2+ 2fy 2=0 {\partial^2 f \over \partial x^2} + {\partial^2 f \over \partial y^2} = 0 Elliptic equations often describe static equilibrium.
  • Hyperbolic: here the classic example is the wave equation 2fx 2 2ft 2=0 {\partial^2 f \over \partial x^2} - {\partial^2 f \over \partial t^2} = 0 Hyperbolic equations often describe waves.
  • Parabolic: here the classic examples are the heat equation ft 2fx 2=0 {\partial f \over \partial t} - {\partial^2 f \over \partial x^2} = 0 and Schrödinger’s equation ft+i 2fx 2=0 {\partial f \over \partial t} + i {\partial^2 f \over \partial x^2} = 0 Parabolic equations often describe diffusion.

The techniques for dealing with the three kinds are very different. They have completely different personalities. Elliptic PDE are the easiest to prove lots of powerful results about, in part because “elliptic regularity” guarantees that solutions are smooth.

To see if a linear PDE is elliptic, you write it down like this: ((4+sinx) 4x 4+ 4y 4x 2x)f=0((4 + \sin x){\partial^4 \over \partial x^4}+ {\partial^4 \over \partial y^4} - x^2 {\partial \over \partial x}) f = 0 and peel off the differential operator involved: (4+sinx) 4x 4+ 4y 4x 2x(4 + \sin x){\partial^4 \over \partial x^4} + {\partial^4 \over \partial y^4} - x^2 {\partial \over \partial x} Then you replace the partial derivatives x,y {\partial \over \partial x}, {\partial \over \partial y} by new variables, say p x,p y. p_x, p_y . You get a function called the symbol of your PDE: (4+sinx)p x 4+p y 4x 2p x(4 + \sin x)p_x^4 + p_y^4 - x^2 p_x The order of your PDE is the highest number of partial derivatives that show up. In the example above, the order is 4.

To see if your PDE is elliptic, just look at what happens to its symbol as the vector p=(p x,p y)p = (p_x,p_y) goes to infinity in any direction. If the symbol always grows roughly like |p| k|p|^k where kk is the order of your PDE, your PDE is elliptic.

In the example above, the symbol indeed grows like |p| 4|p|^4 as pp goes to infinity in any direction. So, it’s an elliptic PDE. So, any solution is automatically smooth: it has partial derivatives of arbitrarily high order!

If you have some spare time, you might convince yourself that the wave equation, heat equation and Schrödinger’s equation are not elliptic. In these examples the order is 2, but there are some directions where the symbol does not grow like |p| 2|p|^2.

(I’ve given examples of PDE involving just two variables, xx and yy. Everything I said works for more variables, too.)

In my youth I mainly did hyperbolic PDE, since I liked physics and especially waves. I looked down on the elliptic folks for working on static phenomena like bubbles and taut drumheads - no life, just sitting there, perfectly smooth. But, elliptic PDE certainly have their charm.

Posted by: John Baez on October 29, 2006 7:57 AM | Permalink | Reply to this

Re: Knowledge of the Reasoned Fact

Thanks a lot, John, for this beautiful explanation. ( I suppose your “cause” for the many derivatives phenomenon will satisfy David.) Moreover, I wish students who have the impression that complex analysis is a piece of heaven while PDE is down to earth “applied” stuff will read it.

Posted by: Gina on October 30, 2006 11:32 PM | Permalink | Reply to this

Re: Knowledge of the Reasoned Fact

I would conjecture that children usually understand why sum and product are commutative and the meaning of substractions but the algorithms for arithmetic operations on 2-digit numbers obscure this understanding. (This was the point behind 42-19.)

Children “understand” that 2+3 = 3+2, because they can see it on their fingers and the notion of the order they put their fingers up in is out of mind. “How could it be any other way?” they think. What I’m not so sure they get is a truly general notion of number, and that “addition” is something abstracted from disjoint union of sets.

Okay, I don’t think they think in those terms. What I mean is that numbers to them are purely instantiated in collections of objects. If they don’t have a specific collection of 32 things to think of as representing “32”, they just don’t think about “32”. The biggest hurdle is to get the student beyond instantiating one number in a set (hold up two fingers) and then another in another set (hold up three more) and counting the result (five fingers). From what I’ve seen, both in talking to the few kids I come into contact with and in talking to adults long past these subjects, almost half the time the best we manage it to teach the decimal algorithms for addition and any real meaning of the numbers is lost. At this point, a number is its decimal expansion, rather than any abstract notion at all.

Posted by: John Armstrong on October 26, 2006 3:00 AM | Permalink | Reply to this

Commutativity.

1. 5+8 = 8+5
2. 6*5 =5*6
4. why a^b is not equal b^a ?

I think that already commutativity of multiplication is subtler than commutativity of addition, so it’s not so surprising that commutativity of exponentiation breaks down entirely.

A + B means (the number of ways) to either choose “left” and do (pick a number less than) A or choose “right” and do B. To see that A + B is equivalent to B + A, you are simply (quite literally in my formulation!) swapping left and right.

A × B means to do A and then do B. To see that A × B is equivalent to B × A, you have to swap before and after, which is not always possible. In this case it is, but only because A and B are independent (both given before your activity begins). In a:AB a \sum _ { a : A } B _ a (of which A × B is a special case), no commutativity is possible in general, since in general B now depends on A.

While a:AB a \sum _ { a : A } B _ a means to do A and then do the appropriate version of B, a:AB a \prod _ { a : A } B _ a means to wait for me to do A and then do an appropriate version of B yourself. So even when B is independent of A (as in AB —whoops, I mean BA!), there is no reason to suspect commutativity, since you’re not even doing the same thing.

So we move from swapping left and right (easy), to swapping before and after (possible in the independent case, but not in general), to swapping input and output (impossible).

Posted by: Toby Bartels on October 27, 2006 12:15 AM | Permalink | Reply to this

Re: Commutativity.

I think these interpretations of the “meaning” of arithmetic operations are interesting, and definitely worthy of consideration. I also think that they move even further from the question, which is on how students learning these arithmetic operations think about them.

Posted by: John Armstrong on October 27, 2006 1:02 AM | Permalink | Reply to this

Re: Commutativity.

John Armstrong wrote in reply to me:

[Your post] move[s] even further from the question, which is on how students learning these arithmetic operations think about them.

Is that the topic of this thread? I had taken the topic to be causality in mathematics, but I guess that it’s for David to decide. (To be sure, Gina considered that topic in the context of how students think.)

The New Math curriculum was supposed to modernise mathematics teaching by bringing in set theory from the early stages, but I don’t think that this was done seriously. However, arithmetic operations on natural numbers (that is, whole numbers, including zero) certainly could be taught along these lines.

Posted by: Toby Bartels on October 27, 2006 3:10 AM | Permalink | Reply to this

Re: Commutativity.

Not the thread as a whole, no, but Gina’s point in raising those particular examples was to suggest a parallel to Chimskian linguistics: ask children learning the concepts “why” they are true and that will give you insight into their epistemic causality. The idea is that the first stabs of an unsophisticated observer towards an explanation contain a deep insight into how the human mind processes the concepts. This is to be contrasted with the viewpoint of an expert who has already thought long and hard about the nature of the subject, and who cannot simply “unknow” that knowledge.

Posted by: John Armstrong on October 27, 2006 3:50 AM | Permalink | Reply to this

Re: Commutativity.

My idea for this thread was to see whether a ‘realist’ notion of something quasi-causal going on in mathematics might be made to work. In that such quasi-causal accounts often need centuries of effort to uncover, as have their physical counterparts, I wasn’t thinking that we’d make contact with children’s early encounters with arithmetic, even if we happen to talk about addition and multiplication, as Toby does.

Having said this, in that the causal accounts strip down concepts to their bare bones (ur-concepts?), it is possible that children’s modes of thinking might make contact with aspects of them. Nevertheless, my own interest is in the long-term disciplinary quest.

Posted by: David Corfield on October 28, 2006 9:15 PM | Permalink | Reply to this

Re: Commutativity.

I learned the New Math in grade school, so I really learned commutativity of addition and multiplication in terms of natural isomorphisms between sets: STTS S \sqcup T \cong T \sqcup S and S×TT×S S \times T \cong T \times S Of course they didn’t talk about “natural isomorphisms” - they just showed pictures of how it works. It’s very obvious stuff (which I am not explaining here).

The New Math may not have helped everyone, but it helped me. Basically they undid the mistake of decategorifying arithmetic. This helped me see that basic arithmetic was about sets of things, not just abstract “numbers”.

I sometimes wonder if, much later, this helped me understand categorification.

Posted by: John Baez on October 29, 2006 8:41 AM | Permalink | Reply to this

Re: Commutativity.

The New Math foundered on many shoals, but its heart was in the right place.

Posted by: John Armstrong on October 29, 2006 1:21 PM | Permalink | Reply to this

Re: Knowledge of the Reasoned Fact

I’m stuck out in the wilds of cyberspace at the moment with the slowest connection known to man, so am finding it hard to keep up with cafe news. There’s something attractive I think to the epistemic view of causality in that it gives an explanation for why mathematical and scientific reasoning share so many features. The question that intrigues me is whether there is a ‘best’ way of organising a field. Even if many accounts were needed for pedagogical purposes, this wouldn’t rule out the notion. It is possible that the understanding of the ‘best’ organisation is only available to one who has worked very hard at a field for many years, acquiring skill and understanding on the way.

To give an example, one would hardly teach a first-year student about adjoint functors to explain why the underlying set of the product of groups is isomorphic to the product of underlying sets, and yet I find it plausible to think that whichever direction mathematics takes, so long as it doesn’t degenerate, it will understand that right adjoints preserving products is at stake, even if this is seen as only a small part of a larger picture.

Posted by: David Corfield on October 26, 2006 10:38 AM | Permalink | Reply to this
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