Postmodern views of math constitute another front on traditional, left-brain-friendly mathematics.
Consider, for example, the following conversation at education.change.org, which I felt compelled to join:
One of the reasons teachers have picked up on my "real world math" lesson ideas is that they teach students that math is a series of functions based on agreed upon rules. Two apples only equal two other apples if we agree that many things either matter or do not matter. Or would you trade two dollar bills for two 100 dollar bills? Or do we, for your sake, average all students and treat them as if they are the same. (why isn't a plate appearance an at bat? why isn't a person without a job considered unemployed? why do we think of certain kinds of numbers as different from others?)
Have you asked a mathematician whether they think their field is culturally constructed? I'm willing to bet that you'd find few mathematicians, any where in the world, who would say that algebra, geometry, calculus, differential equations, topology, or applied math (as in physics or engineering) is culturally constructed--except in the trivial sense that math is the product of culture. Math education is another matter entirely; if math education were less culturally constructed, and more mathematical, it would be a better thing for students of *all* cultures.
Actually Katharine, the mathematicians I know all understand the cultural construct and the sense of "rules." Once you get past 'high school math' this all becomes obvious. No mathematical system works without a shared understanding of its rules.
Whether you read Kline from 1953 or Lave from 1988 or anyone before or since, you'll get the picture.
Let's take a very simple notion - the "prime number." What makes it "prime"? Yes, you've got the answer, but that requires you to believe that, say, the number "1.3" is somehow fundamentally different than 1.0, 3.0, 13.0, or 130.0. Which is not "a fact" but a cultural choice.
This is why any math major knows that there are different geometries, etc, depending on the choice of accepted rules. We are not really stuck with Newtonian Physics, you know.
If prime numbers are socially constructed, so is the speed of light. Your argument applies equally to this concept. What makes it "the speed of light"? Yes, you've got the answer, but that requires you to believe that say, the speed of 299,792,458 is somehow fundamentally different from, say, 200,000,000 meters per second, or 400,000,000 meters per second. Which is not a "fact" but a cultural choice. (For a much more sophisticated argument on the social constructedness of the speed of light and other physical concepts, see Sokal's Transgressing the Boundaries).
As a computer science graduate I can tell you with certainty that at an upper-level math elective, you are taught right away that the foundation of grade school math is based on assumptions. I remember the first time we did a proof dealing with polar math that proved everything we learned about geometry in grade school is simply based in baloney.
The basis of all geometry is the rule that for any two points, you can create two parallel lines. When you get to a high-level mathematics curriculum, You are challenged to prove this and it is simply not possible to prove. It is just an agreed upon assumption so that we can teach geometry.
I may not necessarily see it as a cultural difference, but mathematics is taught in a context that is vastly different dependent upon who teaches it or where you are in the world. Values hold constant, but numbers do not.
What you're calling "assumptions" mathematicians would call "axioms". Euclidean geometry uses one set of axioms; Lovbachevskian geometry (where there are no parallel lines) uses another. Mathematicians use both systems; there's no contradiction or baloney. I learned both systems in 9th grade geometry, and was intrigued by their conceptual coexistence. It's actually rather beautiful. Ask a mathematician!
Math as an abstract idea may be culture-free, but we don't teach it or test it that way. We test math with the expectation that you have memorized and internalized definitions and rules that we have deemed "age-appropriate".
If you see the question "what is 2+2" on a test, you are expected to not think about anything real, but to answer based on the abstraction of a linear counting system. You are not supposed to imagine about what you are counting (that might be apples and oranges), or if perhaps you are counting non-linear events (like 2 laps plus 2 laps around the track get you exactly nowhere, which could be represented as zero.) But because we teach 2+2=4, we expect students to give us back that very simplistic answer on the test.
Students with any imagination often give "wrong" answers for interesting and potentially correct reasons. We are testing the compliance of the student to accept the teaching, not math.
If you asked mathematicians what 2+2 is, you would get a range of answers, questions, and demands for more clarification. It's hardly cut and dry. I can absolutely guarantee that NO mathematician would answer "4" without qualifying the answer with additional information.
People think that because math is logical, that implies that there is always a "correct" answer for every situation. Which if you think about it, is a very Western way to approach things.
Re 2+2, please provide the names of the mathematicians who would not simply answer "4". I know many mathematicians, and I'm curious whether the ones you say would give a "range of answers, questions, and demands for clarification" are people that I know. I can absolutely guarantee that MANY of the mathematicians I know would simply answer 4; but I have not surveyed all of them.
Katharine, you keep asking. Each time you've asked I've not just reached for my bookshelf but put "culture and mathematics" into Google Scholar, pulling up tens of thousands of articles on this issue - most by, yes mathematicians.
It is nothing new. I have a friend who wrote his 1962 Masters Thesis in Math on this subject, and I know it is spoken of constantly in our math education program (which is tied to a 'fairly reputable' mathematics department).
I'd again encourage you to read Morris Kline's work - going back to the early 50s - or to look at the "NYU School" exploring the "fictions of mathematics." I think you will enjoy the conversations.
Ah, but have you searched "culture and mathematics" & "2+2=4"? That's specifically what I was asking about...
Also, there may be a bit of a selection bias in the theses of the articles that turn up under your search... I know many mathematicians (algebraists, analysts, number theorists), but I'm guessing that not a single one who would say that mathematics is, in any way that is deeply significant to math itself, a cultural construct.
PJ Davis in Mathematics (1988) provides for you: http://www.people.ex.ac.uk/PErnest/pome22/Davis%20%20Applied%20Mathematics%20as%20Social%20.doc:
Take any statement of mathematics such as ‘two plus two equals four', or any more advanced statement. The common view is that such a statement is perfect in its precision and in its truth, is absolute in its objectivity, is universally interpretable, is eternally valid and expresses something that must be true in this world and in all possible worlds. What is mathematical is certain. This view, as it relates, for example, to the history of art and the utilization of mathematical perspective has been expressed by Sir Kenneth Clark ("Landscape into Art"): "The Florentines demanded more than an empirical or intuitive rendering of space. They demanded that art should be concerned with certezza, not with opinioni. Certezza can be established by mathematics.Other "open" readings on the same issues
The view that mathematics represents a timeless ideal of absolute truth and objectivity and is even of nearly divine origin is often called Platonist. It conflicts with the obvious fact that we humans have invented or discovered mathematics, that we have installed mathematics in a variety of places both in the arrangements of our daily lives and in our attempts to understand the physical world. In most cases, we can point to the individuals who did the inventing or made the discovery or the installation, citing names and dates. Platonism conflicts with the fact that mathematical applications are often conventional in the sense that mathematizations other than the ones installed are quite feasible (e.g., the decimal system). The applications are of ten gratuitous, in the sense that humans can and have lived out their lives without them (e.g., insurance or gambling schemes). They are provisional in the sense that alternative schemes are often installed which are claimed to do a better job. (Examples range all the way from tax legislation to Newtonian mechanics.) Opposed to the Platonic view is the view that a mathematical experience combines the external world with our interpretation of it, via the particular structure of our brains and senses, and through our interaction with one another as communicating, reasoning beings organized into social groups.
The perception of mathematics as quasi-divine prevents us from seeing that we are surrounded by mathematics because we have extracted it out of unintellectualized space, quantity, pattern, arrangement, sequential order, change, and that as a consequence, mathematics has become a major modality by which we express our ideas about these matters. The conflicting views, as to whether mathematics exists independently of humans or whether it is a human phenomenon, and the emphasis that tradition has placed on the former view, leads us to shy away from studying the processes of mathematization, to shy away from asking embarrassing questions about this process: how do we install the mathematizations, why do we install them, what are they doing for us or to us, do we need them, do we want them, on what basis do we justify them. But the discussion of such questions is becoming increasingly important as the mathematical vision transforms our world, often in unforeseen ways, as it both sustains and binds us in its steady and unconscious operation. Mathematics creates a reality that characterize our age.
And I'd urge you to go the the library, and if you are resisting Kline, read Reuben Hersh's "What is Mathematics, Really?" (1997)
But he doesn't actually deny that 2+2=4. Find me a math professor (a mathematician, not a philosopher!) who says, specifically, that 2+2 doesn't equal 4 in all possible universes, and I'll eat my hat.
I'm a big fan of foundational problems (computation theory; model theory; incompleteness the unprovability of certain mathematical statements--have you taken courses in any of these?) And I'm all for openness, as long as its meaningful!
Kline and Hersh both say this. Read, it'll give you something to chew on.
Believe it or not, sometimes 2 + 2 does not equal 4. It depends on what type of measurement scale you are using. There are four types ofmeasurement scales - nominal, ordinal, interval, and ratio. Only in the last two categories does 2 + 2 = 4.Katharine Beals:
Thus, even on number lines, 2+2=4 is only sometimes true. You need a formal set of, yes, culturally applied, rules to make 2+2=4.
I understand the desire to have something absolute and "always true" in the world. And I am sure it is stunningly frustrating for a pure rationalist to argue with a post-modernist like me, but I challenge you to dig deeper into this.
As Bain's book demonstrates, the shattering of the knowledge system created by K-12 math and science teachers is the first task of professors at top universities.
Apples and oranges! "Nominal scales" use a different definition of number from that used in the statement "2 + 2 = 4"
From the paragraph that follows the one you cite from:
Each number merely represents a category or individual. For example, numbers on baseball or football uniforms are only nominal. Having the number "1" on your uniform does not necessarily mean you are"numero uno" (the best) in your sport. Social security numbers are also nominal. All they do is name or classify the individual.In making statements like 2+2=4, people are not referring to numbers on uniforms. If they were, then the statement would be no more meaningful than "too plus too equals for"!
Speaking of postmodermism and math, have you read Sokol's Fashionable Nonsense?
Wait! A student would have to know your definition of "number" in order to answer that question? I thought this was "absolute" and "culture free"?
But no, I don't read intentionally fraudulent academic writing. I have other things to do with my time.
I've taught math for a number of years, to a number of different students from different cultural backgrounsd, and have *never* encountered a student who needed to know my definition of number (i.e., that I wasn't referring to numbers on uniforms) in order to answer 2 + 2 = 4.
Ira says that 2+2 doesn't always have to equal four: it's a qualified truth.
Katherine wants Ira to find a math professor who will stipulate to the idea that 2+2 does not equal 4 (which doesn't strike me as quite what Ira said). And Katherine says that if Ira finds such a math professor, it proves only that the math professor is a closet philosopher. If I were Ira, I wouldn't spend much time looking...
Without investing the time to read Ira's citations (I have a life), I thought he was pretty convincing. And I thought Katherine was circular: all mathematicians agree with her; if they don't, they're not pure mathematicians.
I think we've all agreed that education (its value in a society, its pedagogy, etc.) is culturally defined. The math discussion has been entertaining; I just can't decide if it has a point in the context of Jon's article.
I will say this: the ability of pure mathematicians to articulate great truths abstractly (which I take to mean hypothetically, in the absence of any real context) is something that I see as a cultural exercise in itself. Most great truths can be articulated concretely or abstractly. You can talk about materialism and the nature of reality or you can talk about Plato's Cave. You can talk about Grace or about the Prodigal Son. Some cultures prefer the concrete presentation. Most Western European cultures prefer abstraction. And that preference is in itself cultural. Even for mathematicians, I think...
To clarify, what Ira said was:
If you asked mathematicians what 2+2 is, you would get a range of answers, questions, and demands for more clarification. It's hardly cut and dry. I can absolutely guarantee that NO mathematician would answer "4" without qualifying the answer with additional information.I then said:
Find me a math professor (a mathemtician, not a philosopher!) who says, specifically, that 2+2 doesn't equal 4 in all possible universes, and I'll eat my hat.Note that Ira is making a claim about what "NO mathematician would do;" I'm asking him to find ANY mathematician who believes that 2+2 doesn't equal 4 in ANY possible universe.
You're also reading too much into what I said about philosophers. There are some really good mathematician/philosophers out there.
So you'd take an answer from a mathametician-philosopher?
Yes. The question, again, is whether the statement "2+2=4", when it is a mathematically well-defined rather than a mathematically nonsensical statement (which was implicit in my initial question but which I now feel the need to state explicitely!), is true in all possible universes.
I'm still waiting for an answer from Ira, or Greg, or Sidney.
But, since, like Greg, I have a life, I've moved on...