Mathematics of Science by Robert MacDuff

Rob,

I think many if not most mathematicians consider pedagogical reform one of the most critical issues in mathematics today. Maybe we don't need to start with noumena and phenomena, but we certainly do need to see the fun in mathematics before it's beaten out of us by fourth grade.

And I think that there may be something inherent in our natures, as Kant thought, that 'gets' it -- without being beaten and stressed. As my then six-year old granddaughter told her mother, "If all numbers go on and on forever without stopping, then all numbers are small numbers." Leibniz would have been proud, and so would Hermann Weyl -- your reference is my second favorite Weyl product (the first is *The Continuum.*).

Hope you get a chance to visit my essay. Yours gets my highest mark and best wishes for success.

Best,

Tom

Hi Rob,

Your reference to your personal blog ("What is one quarter plus one quarter?") got me thinking -- how often have those feigning intellectual superiority compared another's "crackpot thinking" to a quest for the impossible construction "squaring the circle?"

It's as if they deem empirically-based technical activities -- such as applying the compass and straightedge, and measuring the result -- to be the only 'useful' purpose of mathematics, and all other mathematics merely recreational.

It may never occur to them that areas of squares and circles as near equal as measurement precision can arbitrarily approach.

We don't have schools of "Maths and Sciences" comparable to "Arts and Sciences" because there is apparently no art left in the math curriculum that can't be embodied in a practical instrument. I'm led to believe that the stylish new philosophy of "embodied cognition" owes its genesis to Kant, and I don't think that gives Kant much credit for subtlety.

There is a correlation, in my opinion, between the disappearance of arts program -- graphic arts, drama, music -- to the attitude of many "science enthusiasts" that the only math that belongs in the curriculum is identical to embodied cognitive skills. If a research mathematician doesn't call bullshit on that, what research -- what theorem-proving -- can she be doing? True story: as a college freshman, I was compelled to take a remedial no-credit course in plane geometry. My girlfriend, a graphic artist, was in the same course -- same instructor, different time -- so we studied together (sort of, as we probably spent more time studying one another). She, whose life was consumed with geometric patterns, passed with a C, while I got a B. Long afterward, I figured out that while I have no talent for graphic art, I already understood the framework of theorem-proving through years of studying philosophy, and theorem proving was the basis of that course.

So I think there is no real disconnect between your "Mathematics of Science" and Hestenes's "Science of Mathematics."

Kant may be the subtlest philosopher who ever lived.

Lead on, MacDuff! (and don't try to tell me that you've heard that before. )

All best,

Tom

Rob, Painleve' is credited with saying, "The shortest path between two truths in the real domain passes through the complex domain."

I'm very much an admirer of Brouwer (and Weierstrass, Dedekind, Weyl) -- ("all real functions of a real valued variable are continuous") -- as well as Hilbert. I like the Formalists and I like the Intuitionists equally, I think, for different reasons.

Hilbert makes me feel secure about specifying boundary conditions for complex functions in a space I can understand, if not physically experience. Brouwer's "twoity" of functions, and every mathematical act as a "move of time", give me permission to superpose myself, so to speak, within the world of objects as an element in relation. Equality, transitivity, reflexivity become palpable -- so would I like to do logic the way I do algebra? I think I already do. Formal logic, after all, is only a branch of mathematical theory; it is not a theorem in the mathematical canon, such as the fundamental theorem of algebra. I am thinking of that closing line by Doc Brown to Marty in *Back to the Future*: "Roads? Where we're going we don't need roads."

Logic only gets one as far as the dirt path. After that come the graders and pavers, and maybe later even time machine makers. At the beginning of proving a theorem, though, one must take risks and make leaps. Barry Mazur (*Imagining Numbers*) said that when he is introduced to a new result, his initial reaction is often, "I didn't know you could do that!"

So why couldn't any 8-year old be as free as a research mathematician to take risks, make leaps, have fun, be wrong? One may then see mathematics as not so different from "real life." Work and recreation united. I agree with you about the loss of human potential.

Well, here you compliment me for posts, and then I go rambling on. I delight in your posts, too.

Best,

Tom

Any reader who has appreciated Rob MacDuff's essay might want to also read "Modeling the Physical World with Common Sense and Mathematics" by David Hestenes, elsewhere on this board.

My new post to Hestenes' essay contains some further reflections on Rob's work and discusses a deeper connection that I see between the work of the two authors.

Anonymous is me.

Dear Rob,

It seems to me that the issues you cover in your essay could well fill a whole book! I do have one question, though, and a definitive answer to it would go a long way towards illuminating questions about the merits of ideas like the mathematical universe hypothesis (MUH).

I am not a proponent of MUH, but when I tried to formulate a counterargument based on the fact that physics deals mostly with dimensionful quantities while math mostly does not, I realized that there is something that seems to undermine my own position:

In measure theory, one can define measures like length, mass, time etc. even unphysical qualities, and by an argument I read in Cristi Stoica's essay one way to characterize a quantity is to list all possible propositions one can make about it. So, if from a purely mathematical perspective I add units that I symbolize by say "m" to some measure and then append a (possibly countably infinite) list of propositions about "m", then does it not seem that I have in some sense bridged the gap from mathematics to physics?

I would also like to mention that your example in figure 1 reminded of an example given in Yafet Sanchez's essay. His example was that two drops of water coming together to form a larger drop would be considered by us as an inappropriate referent of 1+1=2, but if the world in which we existed had exclusively these sorts of physical features, then we would naturally come to view 1+1=2 as, in a sense, just a meaningless string of symbols because it would then have no referents.

The mention in your bio of your efforts to develop conceptual tools to aid in learning math also reminded me of Sophia Magnusdottir's essay which I just read. The idea she proposes in the second half of her essay struck me like an epiphany, namely that a particular mathematical expression used in a physics context in and of itself is not a model of some observation of reality, but rather the process of using or manipulating that expression to arrive at a conclusion, which by an abstract analogy mimics the actual observation.

I hope you will offer an answer to the question I posed and found my references to the other essays useful.

Best wishes,

Armin

Armin you pose a very interesting question, however I believe the answer is much simpler than you would expect. First of all we are limited to only being able to describe "this" in terms of "that". And this is only possible if there is something defining a connection between the two.

In the case of like (the defining feature determined by us) quantities the connection between them is symbolized by a numeral. In other words numbers are ratios! What is well known in physics (not necessarily mathematics) is that ratios do not add. So I would like to ask what do you mean by 1 1 = 2? Suppose in your example the 1 refers to a drop of water. Then I am curious as to what you believe 1 x 1 might be. It is the answer to this latter question, which I believe is the answer to your original question.

I will have to read Sophia Magnusdottir's essay as it sounds interesting.

Cheers

Rob

Dear Rob,

Without thinking about it too deeply I would take the "meaning" of 1x1 to be any member of a set of propositions which can only be narrowed down by giving further information, because the further information determines the context in which I am using it.

If I understand correctly, one of your arguments can be paraphrased by saying that in mathematics this additional information is usually ignored by replacing the referent with the relationship it has in the ordering to other numbers, effectively representing just itself, whereas in physics it is inevitable that numbers do not just represent themselves but things and relationships between things in the world. Furthermore, this can get confusing if one does not recognize the context in which a number is used (such as pi in your example). I tend to agree with all this.

What is not so clear to me yet is how this answers the question I originally asked. In particular, one could imagine a very long list of propositions in which each describes, say, "m" in terms something else, say, "E", and include in the list all of the proposition that permit one to circle back from "E" to "m", and do this for every single "that" in terms of which "m" is described, and furthermore carefully distinguish whether in each instance a number is used as a ratio or in a purely mathematical sense. That would make the list a giant tautology, but does this really undermine the MUH? If yes, how?

Thanks,

Armin

Dear Rob

Thank you for writing such a thought-provoking essay.

I had the pleasure of having dinner with David Hestenes last week and we discussed your essay (along with ours) at some length.

You have convinced me that multiplication as repeated addition is not the same as multiplication of scale or ratios. While different arguments will have different effects on people, your statement that in the problem

2 2 2 = 3 x 2

the 3 on the RHS acts as an adjective and the 2 on the RHS acts as a noun, really struck me. I get it! Very subtle! Bravo!

Now, you have read my essay, so this may make some sense to you:

I *know* that you can show that the x operator above is associative and commutative which leads to it being an invertible transform of additivity. However, you can also show that x distributes over in repeated addition, which constrains the quantification to being a log so that x must be multiplication.

Now what I bet you can show, is that the symmetries of ratios also lead to multiplication. So I am willing to bet that both problems "repeated addition" and "scaling" are quantified by the same function but for different reasons.

I am going to look into this as that would be really cool!

I also wanted to say that your circle-square problem is mesmerizing and I think that it effectively highlights the fact that there are some unrecognized subtleties still lingering in the metaphors that lead to the mathematics that we use (see Hestenes' essay and mine).

Thank you for a very enjoyable and thought-provoking essay!

Kevin Knuth

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