Strange as it may sound, the power of mathematics rests on its evasion of all unnecessary thought and on its wonderful saving of mental operations.- Ernst MachPhysics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover.- Bertrand Russell
These two quotes were amongst many presented to my theory development class at the start of the term, and when I read them I thought, "Yeah, that's so true..." I thought they were great quotes. But I didn't really get them -- really *get* them -- until yesterday.
The lecture was on thinking and representation, and it included discussion of concepts and how we label them with words which are separate from the concepts themselves. We discussed a paper about concept maps as diagrammatic representations of concepts and the relationships between them: how Y changes as a result of a change in X (e.g., if X increases, Y inceases as well, depicted by a + on the line from X to Y).Though useful, this type of model has at least one obvious limitation: it can't represent precisely how Y changes as a function of X. Many concepts, such as motivation or productivity, are not easy to quantify meaningfully. However, if we are able to measure X and then measure Y and we observe some kind of pattern as we vary them, we can apply a mathematical model instead. My prof used the example of Newton's Second Law, F=ma, and went on to talk about why it's a "perfect model", including the way it's quantitative, testable, generalizable, etc.
But then one student interjected and asked whether it really is such a great model -- does it really serve to convey the relationships between the concepts it relates? He elaborated to the effect of, sure, he "knows" that "force equals mass times acceleration" and can do the related computations, but conceptually, he doesn't *get* the relationship between these concepts from this equation: the prof agreed and went on to ask, "What does it even *mean* to multiply mass by acceleration? Or, if we want to determine acceleration, to divide force by mass??" Conceptually, how do these concepts relate to one another?
Here I suggested that this is where cognition comes into play: as humans, we obviously have cognitive limitations and can only represent so much and manipulate those representations so much before we run out of usable memory and processing ability. For me, I said, classical mechanics makes sense at a conceptual level. I have mental representations of force, mass and acceleration, and I can conceptualize the relationship between them independent of words, numbers or variables. But electromagnetism? V=IR? Sure, I can do all the math just as well as for anything else, but even if I can mentally represent (or "wrap my brain around") the concepts of potential, current and resistance per se, hell if I could successfully represent the *relationship* between them in terms of the concepts themselves. To think about the relationship itself, I have to resort to some kind of analogy involving, say, water and pipes.
And this is where I suddenly *got* it, what it means to say that abstraction is a tool -- that mathematics, the "highest form of abstraction in human thinking", is a tool. If I want to move a box from point A to point B, I can pick it up and carry it there. But if I want to move a whole skid of boxes, I need a forklift to help me.
Abstraction allows us to "[separate] the number concept from what [is] being counted" (Bronowski, 1976). As my prof had put it earlier in the lecture, "Back in the day, we'd think about five trees or five sheep. And then one day some guy comes along and says, 'To hell with the trees and sheep. I'm just interested in this concept of five.'"
Math in particular allows us to completely let go of mental representations of the concepts whose relationships we need to consider. If we can tie numbers to certain concepts by measuring them and observe, through scientific experimentation, a pattern that's consistent with an established mathematical relationship, we also don't have to worry about conceptualizing the literal relationship of these concepts to one another: mathematicians build various forklifts and describe for the rest of us where they'll go if we manipulate the controls in certain ways (and even, if they're especially brilliant/lucky, invent entirely new kinds of forklifts that can pick up different kinds of skids, or even objects that aren't on skids). So if I want to know what happens when I increase the mass and acceleration of an object by so much, I can get into my forklift, manipulate the controls according to some instructions that have been shown to drive the forklift from point A to point B, and not be bothered with actual the skid full of boxes until after I've already moved them. Then, I can get out of the forklift and see that, okay, this is how strong of a force I now have. Manipulating the controls has *nothing* to do with picking up the skid and carrying it myself (were it possible) except the result it achieves.
The magnitude of this sent my mind reeling -- how ridiculously powerful a tool math is, given how well our minds are able to represent concepts and the relationships between them (i.e., not very) versus all of the crazy things we're able to do as a result of circumventing these cognitive limitations. Constructing 50-storey buildings that won't collapse on us? Consider even just the chemical and mechanical properties of the metals, the woods, the concretes... let alone how those play into how much of each you'd need and where they ought to go. Here, those quotes from the beginning of the term came to mind:
I have fallen in love with math. I'd say "all over again", after having allowed myself to become and remain entangled in the depths of mathless biology for years... but I didn't know it well enough before to justify calling it love. Infatuation, maybe? But now, finally seeing it more clearly, I realize that I don't understand it, and it's eating me alive.
It finally became obvious to me, in an epiphany aftershock, what math research is. In the past I'd asked several math-student friends of mine what one actually *does* as a grad student in math (compared to physics students who smash things, chemistry students who blow things up, biology students who kill things, or social sciences students who... recruit people for studies). I'd never gotten a satisfactory answer: "You think. You read, and you think. ... ... and you play around with ideas until something fits."
"Um... okay..."
But yes! Just as those quotes before merely sounded true but now so effectively sum up what I finally *get* about mathematics, this now is so obviously the truth... In science one seeks to discover relationships amongst natural things and describe them in terms of simpler such things that have been previously observed and described; so too in mathematics, where one seeks to discover relationships amongst abstract concepts or other relationships and prove them using more fundamental ones that have been proven before. However, whereas scientists' substrate for observation and description exists as concrete objects the natural world, that of mathematicians is represented abstractly inside their own minds. Hence math research is indeed just thinking after all...
But this is what plagues me now: what are the "most fundamental" patterns, relationships, or concepts? How were *they* proven? *Were* they proven? Are some things just accepted as "true" or "given" (or are they assumptions of some sort), and if so, which things?
What are the irreducible elements of math?
The magnitude of this sent my mind reeling -- how ridiculously powerful a tool math is, given how well our minds are able to represent concepts and the relationships between them (i.e., not very) versus all of the crazy things we're able to do as a result of circumventing these cognitive limitations. Constructing 50-storey buildings that won't collapse on us? Consider even just the chemical and mechanical properties of the metals, the woods, the concretes... let alone how those play into how much of each you'd need and where they ought to go. Here, those quotes from the beginning of the term came to mind:
I sat there in class, just basking in my little epiphany and contemplating the implications, completely oblivious to the ongoing class discussion for at least a good five minutes.Strange as it may sound, the power of mathematics rests on its evasion of all unnecessary thought and on its wonderful saving of mental operations.- Ernst MachPhysics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover.- Bertrand Russell
I have fallen in love with math. I'd say "all over again", after having allowed myself to become and remain entangled in the depths of mathless biology for years... but I didn't know it well enough before to justify calling it love. Infatuation, maybe? But now, finally seeing it more clearly, I realize that I don't understand it, and it's eating me alive.
‘‘Mathematics is only patterns... [It] is not just symbols as names for concepts but is a system of relations with logic and reason built into its inner structure." (Feynman, 1975, 1999)
"Um... okay..."
But yes! Just as those quotes before merely sounded true but now so effectively sum up what I finally *get* about mathematics, this now is so obviously the truth... In science one seeks to discover relationships amongst natural things and describe them in terms of simpler such things that have been previously observed and described; so too in mathematics, where one seeks to discover relationships amongst abstract concepts or other relationships and prove them using more fundamental ones that have been proven before. However, whereas scientists' substrate for observation and description exists as concrete objects the natural world, that of mathematicians is represented abstractly inside their own minds. Hence math research is indeed just thinking after all...
But this is what plagues me now: what are the "most fundamental" patterns, relationships, or concepts? How were *they* proven? *Were* they proven? Are some things just accepted as "true" or "given" (or are they assumptions of some sort), and if so, which things?
What are the irreducible elements of math?
[I'm posting a GIGANTIC comment in multiple parts...sorry! -WXF]
ReplyDeleteEasy. We don't prove the most elementary foundations of math. Mathematicians explore the consequences of "WHAT IF these irreducible elements of math are true? What must follow from them?"
And the next day, some of us asks "WHAT IF these OTHER irreducible elements of math are true? What if the irreducible elements from yesterday are no longer here? What must follow from these new irreducible elements?"
These irreducible elements are called *axioms*. For fun and games, we investigate the consequences of *different* collections of axioms. It turns out that one particular collection of axioms is the starting point of most modern mathematicians: the Zermelo-Fraenkel-with-choice axioms.
These axioms all deal with sets, the foundamental building block of mathematics. Naively, a set is just a collection (think boxes) of things, and in turn, those things are themselves sets. The most fundamental set is the empty set---a set with no elements in it (like an empty box). However, as you aptly point out, the "box" imagery are just intuitive interpretations of abstract concepts, which need no concrete interpretation. The "box" imagery merely helps us think about sets.
Important note: The set containing my dog, your cat, and the elephant in the zoo is the same as the set containing your cat, the elephant in the zoo, my dog, and your cat. That is, repetition and order do not matter. *Two sets are identical if they contain the same elements.* Abstractly, if the memebership status of everything in the universe is the same for sets A and B, then A = B. In the above example, the membership status of your cat, my dog, and the elephant in the zoo are all the same for both sets (they *ARE* members of both sets). Similarly, the membership status of the test tube in the lab, the third chair in my room, and the chemistry building are the same for both sets: these objects are all *NOT* members of the two sets. Since everything in the universe has the same membership status for both sets, these two sets are equal.
ReplyDeleteSo here is the Zermelo-Fraenkel collection of "irreducible elements of math" (all of them, literally, are here!). Note that these axioms are usually written in the language of first-order logic, but I've translated them to English for easier reading.
1. Empty.
These exists a set containing nothing.
2. Pair.
Given two sets A and B, there exists a set containing just two* things: A and B.
[* But if A = B, then there is only one thing.]
3. Extensionality.
Given two sets A and B, if for every set Z, Z is a member of A if, and only if, Z is a member of B, then A = B.
4. Union.
Given a set A of sets, there exists a set whose members are precisely the members of the members of A.
5. Specification schema.
Given any first-order sentence f and any set A, there exists a set B whose members are those members of A satisfying f.
6. Foundation.
Given any set A, there exists a member of A that contains no members of A.
7. Collection schema.
If g is a function realized as a first-order sentence, if the domain of g is a set, then the range of g is a set.
8. Power set.
Given a set A, then there exists a set P(A) whose members are precisely the subsets of A.
9. Choice.
Given a set A, there exists a function f from A into the union-set of A such that f(a) is in a for all members a of A.
It's not important to understand all of these. The point is that there *is* a very rigorously defined collection of "irreducible elements" of mathematics. Again, if you have read the first two paragraphs of this comment, this collection need not be proven, since the enterprise of mathematics is merely asking the following question out of curiosity: "*IF* this collection of axioms are true, what truths must follow?" If you don't like this collection of axioms, make your own! And investigate what must follow from those axioms! Your mathematics will differ from that of the majority of mathematicians, but nothing stops you from starting with a different collection of axioms.
ReplyDelete(By the way, set theory is the foundation on which every other branch of mainstream mathematics can be put---that is, all things that can be proven in those branches of mathematics can trace their truths to the Zermelo-Fraenkel axioms).
As an example, suppose we do not assume the Axiom of Choice. Then it turns out that a whole slew of things are still provable from the other axioms of Zermelo-Fraenkel. But it also turns out that certain things are no longer provable. For example, it is no longer true that we can prove Zorn's Lemma or the Well-Ordering Principle.
In summary, yes, there are irreducible elements of mainstream mathematics called the Zermelo-Fraenkel Axioms. But there may be a misunderstanding about the role of such a collection of "irreducible elements of mathematics". For one, these axioms need not be proven (and can't be proven!). These axioms merely form a convenient starting point of mathematics, and can be discarded at any time by the curious. The curious can in fact invent their own collection of axioms, and investigate the truths that follow from them. Most mathematicians use the Zermelo-Fraenkel axioms as a starting point of their work. But indeed, some choose to start elsewhere. In fact, quite a few mathematicians do not include the Axiom of Choice in their starting collection of axioms, but they accept all the other Zermelo-Fraenkel axioms. But it all depends on their mood and the phase of the moon. If we don't say anything, assume we started from the full Zermelo-Fraenkel collection of axioms. It's "Zermelo-Fraenkel unless stated otherwise." Some people publish papers whose whole point is to prove a certain statement without using the Axiom of Choice; in those papers, they will clearly state that they are not starting with the Axiom of Choice (and still manage to prove the statement!).
Hope that helped.