I have some questions! I'm a high school Physics teacher and I recognize in this paper the difference between how my students experience numbers and how I want them to... realizing that for them, their numerical "answer" is an entity unto itself kind of makes it a little bit more understandable how they constantly leave off units or fail to interpret their calculations!

So overall I think I get the concept, but I'm struggling to understand some of the details. Here are my thoughts...

1) On pi as a "pure" or "unitless" number: I think that for scientists, pi actually DOES have units: it's meters of circumference per meter of diameter. When my students find linear relationships between quantities measured in the "same" unit (for example, a bouncing ball's drop height vs. rebound height) I always stress the point that the units don't "cancel out" in the slope, they maintain their meaning. The reason why it's not a "pure" number is because the meaning comes from the relationship between the two quantities, as you state.

2) I wish there was an example, at the end of the same section, of a situation in which the axiom a (b/b) = b (a/b) does not apply. I'm having a hard time with this. And if that axiom is only true when there's a physical law that supports it, then why not just start with pi = C/d, with the reasoning being that we have a law supporting it? To my untrained eye, I don't grok the significance of spelling it out so much in the "proof" section. Any way to explain why this is necessary?

3) I'm not clear on why in lines 15-16 in "does 1=1", are we allowed to substitute 1 for C/C'. Isn't that sort of begging the question if we're trying to get to C = 1 C'? Again I think I must be missing the point.

4) I think that I need an example of a law-like quantitative relationship for the section "Mathematical Operations Encode Relationships" in order to really get it. Could a and b represent positions, for example? In that case what's the "lawlike" relationship that allows us to compare them with a delta quantity?

5) Here was my thought process when I read "It is not possible to divide distance by time, which is sometimes considered to be the case" - let me know if I landed up in the right spot: Why is it not possible to divide distance by time? If I measure that in 3 seconds an object has moved 12 meters, can't I divide the 12 meters into 3 equal groups and assign each group to one of the seconds, stating that it moved 4 meters in each second? And why is that not dividing distance by time? I guess it's because I'm really dividing the distance into three equal groups of displacement, which IS allowed, and assigning each group to a time interval is a separate operation - the structural part. Right? So is it possible to divide ANYTHING "by" another thing? It seems like according to this paper the answer would be no, that's an idea that only makes sense if numbers are reified. The count on the number of resulting groups is a different count than the count on whatever quantity is to be divided.

5) I'm pretty sure the whole vector algebra section is above my pay grade. I don't even know what to ask about this part. But I think I am not the target audience.

Anyway, I think that understanding the concepts in this paper definitely has implications for my goals and methods as a high school Physics teacher - this is a sense of the meaning of mathematics that I work hard to inculcate in my students. So even though my questions probably sound like a kid who missed the first 3/4 of the lecture and demands to be caught up in the last 15 minutes, maybe you could see your way to taking some time to clarify? Thanks!!

Lucas, you are perfectly correct in assuming that many of these ideas will ease instruction. Your comments reflect, to some degree, my own struggles to understand.

1) Numbers are used in three different ways: ordinal/relational, cardinal/quantitative and relationship/structural. In mathematics the latter concept is tacitly used to construct numerical symbols but from then on they are treated as cardinal. My point here is that in science all numbers encode part to whole structure and knowing how symbols encode structure is what makes science understandable. Re: Pi having units. I gather you are saying n meters of diameter. But isn't your n a relationship number as it encodes the diameter/meter connection?

2) There is a tendency to consider physics from a purely experimental basis and ignore the theoretical side. C/d means there is a quotient relationship between C and d and the value of the relationship has to be determined by experiment.

Axioms in math are always assumed to be true. However in science they are contextually true. In other words C can't be multiplied by any fancy form of one (FFOO) and make sense.

3) In math 1 = 1, in science a/a тЙа b/b most of the time. However in a = 1a, the 1 is a placeholder that allows a process for describing a in terms of b. Also what is being shown is that technically the a's are not the same, rather the same but not the same as in this is one of those.

4) Great question, if an object moves from a to b, then the displacement (relationship) is defined by the object. If the object doesn't go through those points then what would it mean?

5) Distance over time is a ratio, so haven't you just constructed an equivalent ratio?

Two quantities have a relative magnitude, two vectors have both a relative magnitude and orientation. By inspection, a = (ab/bb) b has to be true as it can't be otherwise. Let a = 2x + 4y and b = x + y be vectors in two-dimensional space. (ab/bb) = (2x + 4y)( x + y) / (x + y). Multiplying that relationship by b will produce a. I will let you discover the connections between xy and yx; x2 and y2. It will not take anymore than a few minutes to work out the necessary algebra. Play, have fun and enjoy figuring out the relationship one vector has to another.

Cheers

Rob

P.S. Try giving it to your students as a project.

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

Tom,

I totally agree on your astute awareness of the critical issue of pedagogical reform. Already a third of students have been lost by grade 3 and 85% by high school. The destruction of human potential is extreme. The difficulty is that we don't teach math we teach computation. We teach axioms not model construction. We teach operations not concept construction. The question as to what is a number is rarely if ever addressed.

I have low hopes for an integrated math/science program anytime soon.

By the way, I love your reference to mathematics as being a religion.

thanks

Rob

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

    Tom, I am wondering if you have missed the point of the circle/square problem. May I be so bold to say that there is this "religious" (your term) belief in the power of mathematics and yet it is unable to solve such an incredibly simple problem. Once noticed, all of a sudden there is this infinite variety of simple structures current mathematical programs are unable to model.

    Complex numbers with real and imaginary axis may seem delightful toys. The true delight occurs when you can perceive and reason about the real world and the word imaginary slips off into oblivion.

    Let me put it another way, I think that when mathematicians chose the path of rigor over intuitionism (Hilbert vs Brouwer) they threw the baby out with the bath water. Boole did the same to logic. Seriously Tom, would you not like to be able to do logic in exactly the same way that you do algebra? Write out the equations and algebraically deduce the answer. No, I guess not, as that would not be "merely recreational."

    There are many similarities between David and myself; however, there are also significant differences.

    As to Kant, far beyond brilliant, I am always stunned by what he was able to do. Hume as well, and to call him an empiricist is downright cruel. I believe that he is the that lead Kant to the concept of synthetic a priori.

    And Tom your posts are a delight

    cheers

    Rob

    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

    7 days later

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    • Post #15

    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.

    17 days later

    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

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      • Post #18

      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

        Armin you have in fact answered your own question by realizing that the 1x1 does not have any meaning in the context that I proposed. What you will quickly discover that no matter what proposition you construct for 1 1 it will not work for 1x1 or 1/1 or 1-1. These operations represent four different types of structures required for developing a set of symbols to represent structure contained in the world.

        MUH if I understand it correctly is assumes the existence of mathematical structure. However, as Hilbert and others realized, numbers have only one structure and that is a serial order. What is amazing is the vast interpretations of number that arise out of all the possible contexts in which it can be utilized but that doesn't even begin to qualify for all the structures required to construct a world. My paper provides one example, which provides a template for an infinite variety of situations mathematics is incapable of modeling.

        Regarding your last example, yes you may imagine a very long list of propositions. However, what you are missing is that each entry in the sequence of "this" and "that" has to be defined by something. If you decided to construct such a loop then you would be constructing a self-referential loop, upon which you might find that you are your own grandfather.

        Cheers

        Rob

        9 days later

        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

        I am glad you had a wonderful time talking to my good buddy David: such an outstanding individual.

        The hardest thing to grasp is that there really are serious fundamental foundational issues with mathematics. The circle square illustration is just one of thousands. Recognizing that there are issues, is only part the problem, finding ways to illustrate them and solutions to them are others.

        What I tried to point out in this paper is that dropping units may seem to be a logical approach to a generalized mathematics. However, so much is tossed out, requiring weird machinations to make it work. There are three different types of numbers: ordinal, cardinal and relationship. Currently mathematics switches back and forth between these without any indication that it has done so.

        To be able to clearly see the issues requires what I call a "mathematics without numbers". What becomes obvious is that numbers do not encode quantitative information but rather structural information. This then opens a doorway into a world of thinking, reasoning and logical connections. I believe that if Russell, Whitehead, Frege, etc. had discovered this way of thinking about number, math would be vastly different today.

        Rob

        BTW: The difficulty between the three's and two's was I believe (although I can' t find it) pointed out by Hermann Weyl, as a difficulty with set theory.

        2 years later

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        • Post #23

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