## Saturday, September 5, 2009

### Postmodern Math: Does 2 + 2 always = 4?

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:

Ira Socol:

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?)

Katharine Beals:

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.

Ira Socol:

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.

Katharine Beals:

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).

Sidney Andrews:

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.

Katharine Beals:

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!

Sylvia Martinez:

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.

Katharine Beals:

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.

Ira Socol:

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.

Katharine Beals:

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.

Ira Socol:

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.

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.
Other "open" readings on the same issues

http://www.economics.pomona.edu/widner/courses/econ58/ps/whatmath.pdf

http://www.members.tripod.com/~jan_dejnozka/peano_russell_quine_number.pdf

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)

Katharine Beals:

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!

Ira Socol:

Kline and Hersh both say this. Read, it'll give you something to chew on.

and
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.

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.
Katharine Beals:

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?

Ira Socol:

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.

Katharine Beals:

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.

Have you?

Greg Cruey:

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...

Katharine Beals:

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.

Greg Cruey:

Fair enough.

So you'd take an answer from a mathametician-philosopher?

Katharine Beals:

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...

Hainish said...

This is why I don't read that blog very much. Folks there don't have an analytical bone in their collective body.

(Of course, then I go read ktm and have right-wing politics shoved in my face. So it's a trade-off.)

Katharine Beals said...

Hainish, Where do you see right-wing politics at kitchentablemath? I'm very interested in the political/education connection, and would love to hear more of your impressions.

bky said...

This was a very long post and I didn't read it all, but there is both a point and a non-point here.

Is 2 + 2 always 4? If you are talking about integers yes it's 4. Now, wink, wink, nudge, nudge, what if we are talking about integers mod 3, to take an example. Then 2 + 2 is 4 which is the same as 1. So 2 + 2 is 1. Perfectly well-defined since then the object "2" and the operation "+" are defined over a finite group which is not the same as the integers. So yeah, in that sense mathematicians would say 2 + 2 is not always 4. But if the context were elementary education, I think every mathematician would say, uh, 4. Because we are not talking about finite groups. All this is in no way "culturally determined". It is mathematically determined. You are either talking about what you seem to be talking about from context (counting numbers) or you are talking about something that could not be used to count cookies, that is, finite groups.

It is very misleading to say, well, look, it all depends on the rules that you agree on, doesn't it? No, Humpty Effing Dumpty, it all depends on whether you are talking about one specific thing or another specific thing.

Hainish said...

I didn't think the politics were at all subtle, but here's a recent example:

http://kitchentablemath.blogspot.com/2009/09/compare-and-contrast-part-3.html

And this:

http://kitchentablemath.blogspot.com/2009/08/cult-of-personality-as-character-ed.html

And the impression also comes about because of the number of comments expressing right-wing opinions, as well as the links to right-wing blogs. (Right Wing Prof, for example, might ostensibly have a good reason for being linked, but he blogs about politics much more often than anything to do with teaching.)

(And don't get me started on Instructivist.)

Hainish said...

This

and this.

vlorbik said...

"what is 2+2?"
is of course

unless it's implicit.
a four-year old?
one thing maybe.

somebody testing a piece of computer code?
another maybe.

somebody trying to catch you out
in some pseudophilosophy?
maybe still another.

sure and it's empty to posit that
"properly" defined, correct symbol
manipulation yields correct results.

owen by the way

oh... and by the way...
i was created a doctor
of philosophy [for my
sins] in '92 and have
professed maths.

so here goes an experiment.
these SOB's downtown
have decreed that push
these damnable calculators.
let's see what happens when
i put "2+2=4" [the whole string]
into the command line.
wow. 1.
that means true.

okay. you win this one. sort of.

let's see.
pretty interesting.
would't call it "true" though.
well-defined" or some such
weaselword dodge? of course not.
(everybody knows that there are
always *undefined* terms and
other such conventions; this
isn't at all when we're doing math
and seldom when we philosophize).

PS
2+2=5 is the symbol in 1984
of the kind of "truth" that can
only be beaten into you.

mathwarriors split on whether
2+2=4 is of the same type.
i myself do not claim
to know the answer to this.

vlorbik said...

"..isn't at all *interesting* when we do math..."

(near the end of my last post;
word missing.
if anybody got that far....)

congrats on the book
by the way.

PPS
the unionbashing at KTM gets so thick
sometimes i don't come around for weeks.
just sayin.

bky said...

Vlorbik -- back when I was working for a high tech company one of the current management slogans was 1+1=3 or something like that, which was supposed to be a clever statement about synergy or somesuch thing. As in "we are so good that when we put two things together some nonlinearity of our corporate prowess produces more than the sum of the parts!" Of course putting 1+1=3 on a powerpoint slide really looks a whole lot more like saying "we are dingbats."

Lsquared said...

Hm, just read Vlorbik's comments. I agree of course. In support (and with a slightly different slant on it)...

I'm a math professor and a mathemetician. 2+2=4, because 4 _is defined to be_ 1+1+1+1, and 2 _is defined to be_ 1+1, and addition (on any subset of the complex numbers, and in any system where the operation is written "+") is associative.

There are a few mathematicians who would say, yes, it's 4, and would you like to read a long treatise on arithmetic, and why that is so, but there are no mathematicians that would say it is not 4. Now, there are probably a few applied mathematicians and really tired teachers of freshman who would subject you to a 5 minute lecture about how, when you write 2+2 without units, you are implicitly communicating that the units are the same, and that if you choose to write 2+2, you should make sure the units are the same first (units are things like cm and m and pounds and ounces), and ideally you should be including your units if your problem is of an applied nature.

Anyway. I, and almost all of the mathematicians I know, would just answer 4, and would think the rest of this argument is not worth wasting time over.

PhysicistDave said...

Ah, Katharine! You are indeed a patient lady – I fear I would have lost my temper and made some ad hominem remarks with those folks.

On the “politics at ktm” issue, since I inadvertently diverted your thread over on ktm, let me offer a couple thoughts here.

Almost everyone at ktm is dissenting from an approach to teaching math that is widespread in the “public” schools – i.e., schools operated by the government. People who are already skeptical of government will find it easier to be critical of government schools. And, people who have come to dislike the way the government runs the schools may also come to wonder if maybe the government messes up lots of other things, too.

So, if “right-wing” means somewhat skeptical or critical of the government, you would expect to find many (but not necessarily all) of the denizens of ktm to be “right-wing.”

Of course, there are other meanings of “right-wing.” Supporting G. W. Bush’s invasion of Iraq would not demonstrate skepticism towards government, but it was widely labeled “right-wing,” even though a lot of prominent right-wingers (Bob Novak, Pat Buchanan, G. G. Geyer, even the Birch Society) opposed it.

Thanks to Google, it is possible to test in what sense ktm is “right-wing.” There is, as one might suspect, a great deal of criticism of government schools, and a fair amount of criticism of government in the abstract.

However, a bit of Googling shows little discussion of the Iraq War on ktm: in fact, I could not find anything that indicated what anyone thought about the war at all, although a couple of rhetorical asides did seem to suggest the Iraq War as an example of a plan that had not worked out too well (perhaps, that is neither a left-wing nor right-wing perspective, since the war has not exactly worked out as planned).

So, I conclude that, yes, ktm is “right-wing” if “right-wing” means exhibiting some skepticism of government. If “right-wing” means supporting the war in Iraq, etc., well, whatever participants at ktm think on that issue, even with the power of Google, I cannot find those thoughts on display at ktm.

All the best,

Dave Miller in Sacramento

Katharine Beals said...

Vlorbik, thanks for your posts; this one is especially inspiring to read. And thanks for the congrats and for introducing me!

Katharine Beals said...

Dave, I agree with all the connections you draw between education and politics. Barry Garelick's story on Kitchentablemath suggests one more: the more polarized things are, the harder it is to publish criticisms of current practices in non right-wing venues, or to get liberal/democratic politicans to voice them--which, of course, can lead to a particularly vicious cycle.

PhysicistDave said...

Katharine,

Yeah, turning education into a left-right issue actually does not make much sense.

And, of course, some of the prominent people advocating an academically solid curriculum (e.g., E. D. Hirsch) are not political conservatives.

In my observation, this polarization is more due to the left than the right. Remember the big “ebonics” debate in Oakland? I have in-laws in the Oakland area, and they were pooh-poohing critics of “ebonics.” Yet, many of the critics were simply making the point that black kids are not well-served if they fail to learn standard English, just as Asian or Hispanic (or white) kids who live in the US are ill-served if they do not learn standard English: indeed, one of the more articulate critics of teaching ebonics was John McWhorter, an African-American linguist at UC Berkeley (incidentally, McWhorter’s book, “Power of Babel,” is fascinating – although it has nothing to do with ebonics, politics, etc., just pure linguistics).

Yet, because some critics of ebonics were on the “right,” my liberal in-laws reflexively defended it.

Perhaps, in one respect, it does make sense, though. One meaning of “left vs. right” is whether one basically wants to preserve the modes of human interaction, the personal values, etc. that prevailed, say, a hundred years ago, or whether you hate the values and attitudes that prevailed a hundred years ago.

The code-words we all know, “patriarchy,” “alternative life styles,” “feminism,” etc. tend to boil down to that question. Ultra-right people think the country was in pretty good shape a hundred years ago: it just needed air conditioning and modern electronics. Ultra-left folks think almost nothing was right in America a hundred years ago or, perhaps, even fifty years ago.

Of course, most of us are in the middle: we approve of the more sane treatment of gays and the end of Jim Crow, but we are not happy about the higher crime rate, etc.

Logically, those of us arguing for a more academically solid curriculum are not really calling for a return to the past. But we do often sound that way. After all, when we want to point out that the teaching of math could be more rigorous, an easy way to make our point is to show that math teaching once was more rigorous, even forty years ago.

In fact, of course, anyone arguing for a solid curriculum cannot simply advocate a return to the past: typical nineteenth-century curricula, for example, could not have included modern science, simply because most of modern science had not been discovered yet.

But I think it is easy to see how many “cultural leftists” could think that a simple return to past models of education is what we are advocating.

Personally, I’m an atheist and an anarchist, so, logically, I should be on the extreme “left” end of the cultural spectrum. And, I actually do see a solid education as a way of empowering people so that they can reject the lies of religion, government, etc.

And, yet, I do think that some of the old values had merit: perseverance, hard work, personal responsibility, etc. I also am not too happy about violent crime, kids whose lives are destroyed by drugs (although I oppose Drug Prohibition as a failed policy), etc.

So, I tend to find that I get labeled a “right-winger.”

I have occasionally tried strongly emphasizing the point that *I* want kids to be solidly educated so that they can help bring about the grand cultural and political revolution that I favor. It turns out that this is not that appealing to liberals either: most contemporary liberals are, in reality, rather conservative and do not really want a radical change in the status quo.

So, we are probably stuck with this rather strange situation. It’s pretty weird, but kind of amusing.

All the best,

Dave

pomo mathematician said...

Prime numbers would be my first example of a mathematical concept that's NOT socially constructed -- or if it is, then barely so.

The speed of light is another great example of a phenomenon that's NOT socially constructed -- or if so, only in a trivial sense.

When you get to talking about changing assumptions, then I think you're getting into social construction, as Socol rightly points out.

Take for example Bishop Berkeley's famous critiques of calculus -- or Kronecker's critiques of mathematical use of infinity. Some mathematicians feel it's appropriate to use non-constructivist proofs employing the Axiom of Choice,

Beyond all that ... I think there are at least three interesting intersections of postmodernism and mathematics.
(1) Applications of mathematics to postmodern thinking -- making statements like "The topological connection between
(2) Applications of pomo to teaching -- maybe that would mean asking questions or using math to deconstruct assumptions rather than following steps (see Lockhart's Lament)
(3) Questions about whether mathematics or fields of mathematics are socially constructed or display other pomo characteristics.

I think (3) is the least interesting of the options, but I see it as the most-discussed topic.

-I'm reminded of Barry Mazur's "When is One Thing Equal to Some Other Thing?"

-Why are you citing Transgressing the Boundaries?

-Is Socol a parody on Sokal? Is this whole article a parody?

-Why does Google need to supply a result with "culture and mathematics" and "2+2=4" ? Isn't it enough that some mathematician writes about the issue somewhere, using whatever language?

-I think that, postmodernly, what "a mathematician" would answer to 2+2=4 depends on the CONTEXT. If it's during a philosophical debate or where they have reason to guess that you mean something other than "typical, assumed" numbers with typically assumed properties, you would get the interesting answers. It also depends on how deeply the mathematician has thought about this stuff, how hard you press, etc. There was a good article on davidaedwards.tumblr.com about a month ago about the culture of mathematics, the personal obsessions that drive the mathematicians, and their dis-ease at looking outside what questions they need to be asking to prove properties about THEIR particular theoretical object.

pomo mathematician said...

Prime numbers would be my first example of a mathematical concept that's NOT socially constructed -- or if it is, then barely so.

The speed of light is another great example of a phenomenon that's NOT socially constructed -- or if so, only in a trivial sense.

When you get to talking about changing assumptions, then I think you're getting into social construction, as Socol rightly points out.

Take for example Bishop Berkeley's famous critiques of calculus -- or Kronecker's critiques of mathematical use of infinity. Some mathematicians feel it's appropriate to use non-constructivist proofs employing the Axiom of Choice,

Beyond all that ... I think there are at least three interesting intersections of postmodernism and mathematics.
(1) Applications of mathematics to postmodern thinking -- making statements like "The topological connection between
(2) Applications of pomo to teaching -- maybe that would mean asking questions or using math to deconstruct assumptions rather than following steps (see Lockhart's Lament)
(3) Questions about whether mathematics or fields of mathematics are socially constructed or display other pomo characteristics.

I think (3) is the least interesting of the options, but I see it as the most-discussed topic.

pomo mathematician said...

-I'm reminded of Barry Mazur's "When is One Thing Equal to Some Other Thing?"

-Why are you citing Transgressing the Boundaries?

-Is Socol a parody on Sokal? Is this whole article a parody?

-Why does Google need to supply a result with "culture and mathematics" and "2+2=4" ? Isn't it enough that some mathematician writes about the issue somewhere, using whatever language?

-I think that, postmodernly, what "a mathematician" would answer to 2+2=4 depends on the CONTEXT. If it's during a philosophical debate or where they have reason to guess that you mean something other than "typical, assumed" numbers with typically assumed properties, you would get the interesting answers. It also depends on how deeply the mathematician has thought about this stuff, how hard you press, etc. There was a good article on davidaedwards.tumblr.com about a month ago about the culture of mathematics, the personal obsessions that drive the mathematicians, and their dis-ease at looking outside what questions they need to be asking to prove properties about THEIR particular theoretical object.

isomorphismes said...

You should talk to owen.maresh.info. He made this isomorphismes.tumblr.com/post/59550012001/the-swift-luminescent-energy-drink-of-the-psyche and http://math.stackexchange.com/users/890/deoxygerbe lists these views:

"""
I am kind of skeptical of doing much in the way of symbolic nonsensica. The two dimensional symbol manipulation language we use is not itself mathematics (would the Vulcan learning pits in the first Abrams Star Trek movie actually use human symbols for math?) and is culturally and historically contingent. I like special functions and making phase portraits of functions on the complex plane. I don't trust the sort of mathematical writing which solely consists of pages and pages of category theory diagram scribble without alternate metaphor validation.
"""

He's trying to reach out to young kids to give them a physical sensation of super high level mathematics (Schemes, topoi, q-series) through methods other than pointy symbols, such as the didgeridoo. His twitter is @graveolens.

isomorphismes said...

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.

Here are some counter examples.

In the above Eugene Volokh says how setting 6^0 = 1 is a matter of social agreement.

This is elaborated further in http://en.wikipedia.org/wiki/Tychonoff's_theorem where (at least at the time I posted http://tmblr.co/ZdCxIy2zjyTE) said:

"""
Tychonoff’s theorem gives confidence that our definitions of compactness and product topology are the correct (i.e., most useful) ones.
"""

Again this is social agreement or construction. I think you would see something similar if you explore enough different definitions of e.g. entropy. People are searching and discussing what are the right things to define. It's not debatable what the results of a calculation are, but that's not the creative act in mathematics.

Finally and most powerfully: http://arxiv.org/abs/math/9404236 a famous mathematician defines mathematics to be a social phenomenon.

Anonymous said...

They are complicating things aren't they --- kind of like asking philosophically what a number is, who are we to make a number in the first place ... how do we know a number is a number. It strikes me as philosophy and they are skipping math.