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Let’s try to motivate schemes

Recently there has been some controversy over David Mumford’s Nature magazine invited obituary of Alexander Grothendieck being initially rejected on submission (see here and here). At issue was the attempt to explain the mathematical idea of schemes (one of Alexander Grothendieck’s most important contributions) to a non-mathematician audience. Professor Mumford is a mathematician of great stature and his explanation is better than anything I could even attempt. However, in addition to the issues he raises I don’t think he was sensitive enough to what a non-mathematician considers motivation.

I’ll take a quick stab at explaining a very tiny bit of the motivation of schemes. I not sure the kind of chain of analogies argument I am attempting would work in an obituary (or in a short length), so I certainly don’t presume to advise professor Mumford on his obituary of a great mathematician (and person).

A quick warning: I am a Ph.D. computer scientist with an undergraduate education in mathematics (plus some graduate work in mathematics). I have never worked with schemes, but I have worked with computational algebraic geometry. I can’t explain schemes to you, because I frankly find them a bit abstract. But I can explain a near-relative or ancestor: varieties. From that I think I can at least motivate schemes. But again, I am only going to explain the sliver that excites me: so I am going to neglect a lot (describe a very important work as being merely important).

What non-mathematicians often don’t know and mathematicians forget to explain is: the reason mathematics tolerates strange and abstract definitions is to make theorems stronger and simpler. Despite what it seems from the outside, obscurity and strangeness are not valued in mathematics.

Let’s start what is considered a concrete example: the fundamental theorem of algebra. I almost said “let’s start with the complex numbers,” but that is exactly the kind of “cart before the horse” mis-motivation I don’t want to make.

From the Wikipedia: “Peter Rothe, in his book Arithmetica Philosophica (published in 1608), wrote that a polynomial equation of degree n (with real coefficients) may have n solutions.” This is an exciting possibility with tons of applications, people very much wanted this to be true. It would mean you could write any polynomial as product of linear terms, and you could solve a lot of concrete equations and problems. The catch is: when you get precise you find out the statement isn’t true. There is no real number that is a solution to the polynomial equation x^2+1=0.

To fix this you go one of two ways.

  1. You state more complicated, less powerful, and less appealing versions of the theorem that are correct.

    Such as: a polynomial equation of degree n (with real coefficients) may be factored into a product of linear and quadratic terms. Notice it isn’t just the proofs that are getting complex it is also the statements of conditions and results.

    This is undesirable: we were forced to move from solutions (numbers that when plugged into the polynomial simplify the whole thing to zero) to factoring polynomials. And we ended up with two types of terms in the factorization: quadratic and linear terms. The theorem is vacuously true when applied to the polynomial x^2+1 as it just says it factors to itself.

    This assertion was sufficiently complicated that as late as the 1740s mathematicians as notable as Gottfried Wilhelm Leibniz and Nikolaus Bernoulli were (incorrectly) claiming to exhibit polynomials that did not so factor.

  2. You replace your current abstraction (the real numbers) with a new one better suited to encode the theorem you are interested in.

    In this case we introduce the complex numbers (a different number system than the reals). Then the following theorem is true: all polynomial equations of degree n (with complex coefficients) have n complex solutions (counting with repetition). From the Wikipedia again: Gerolamo Cardano introduced the complex numbers around 1545. This is a monumental step in mathematics and by the mid 1750s many attempted proofs of this theorem were published (now all considered to be incomplete in that they assumed a few things not yet known/proven). By 1821 complex number based proofs are making it to textbooks.

So the complex numbers allowed the simplification of strongly coveted theorem and drove hundreds of years of mathematical research.

Let’s move one step closer to schemes: polynomial ideals and affine varieties.

I like polynomial ideals. The reason is: a great number of very hard problems can be encoded as asking if a given polynomial (in possibly more than one variable) is in a special type of set called an polynomial ideal. There are algorithms for working with polynomial ideals (in particular the Groebner basis reduction and the Buchberger’s algorithm). The natural companion object to an polynomial ideal is something called an affine variety. Think of polynomial ideals as special sets of multivariate polynomials and think of affine varieties as sheets of points these polynomials are simultaneously zero on. So polynomial ideas are generalizations of polynomials (to more than one polynomial and more than one variable) and affine varieties are generalizations of solutions (to sets of points describe many different assignments to multiple variables).

This set of mathematical tools and algorithms under research since the mid 1960s translate a lot of the most important algorithms from linear algebra (such as Gaussian elimination) and number theory (such as computing greatest common divisors) into a unified framework over multivariate polynomials. These algorithms are why packages like Macsyma, Maple, Mathematica, and SymPy can solve many equations.

Polynomial ideals and varieties are related in an interesting way: the bigger the polynomial ideal the smaller the corresponding variety. For example all of R^n is a solution to the set of polynomial equal to { 0 }. And only the points { i, -i } are solutions to the single variable polynomial x^2+1. This sort of linkage is called a Galois connection finding theorems like this motivates a lot of category theory. The idea is: we are working directly with polynomial ideals, but the affine varieties (or sets of simultaneous zeros) help us and do a lot of the bookkeeping for us (making it much easier to prove a lot more theorems).

Except, the relation doesn’t quite work as well as we would like. We would like something simpler and more powerful than a mere Galois connection: to have affine varieties be in one to one correspondence with polynomial ideals. That way each one has enough detail to be used as for detailed record keeping on the other. It turns out affine varieties are not quite up to the job. Affine varieties can not carry as much detail as the corresponding polynomial ideals. Affine varieties are only tracking details of a subset of polynomial ideas called radical polynomial ideals. A radical polynomial ideal is such that if p(x)^k is in the polynomial ideal for some integer k ≥ 1 then p(x) is in the polynomial ideal. So the set {x^{2k} | k>=1} is an polynomial ideal, but not a radical polynomial ideal (the corresponding radical polynomial ideal is {x^{k} | k>=1}). A polynomial ideal and its corresponding radical polynomial ideal are associated with the same affine variety (so the space of affine varieties can’t tell them apart).

The issue is: we wanted to work directly with polynomial ideas (we had some great problems and algorithms ready to go). Affine varieties were only introduced to help with the record keeping (mostly in proofs). We don’t want to mess up our work by switching from polynomial ideals (that encode what we want) to radical polynomial ideals (which add in more constraints). What if instead of fixing the polynomial ideals, we fixed the affine varieties? Affine varieties are the ones not doing their job. It turns out we can in fact work with polynomial ideals: we just need to replace affine varieties with an more detailed abstract structure called schemes.

If you want to work with general ideals (that is subsets of arbitrary rings closed (and absorbing) under multiplication, not just ideals of polynomials) then your natural most detailed “sets of solutions” structure is not varieties but schemes. Alexander Grothendieck in worked this out in the 1960. For some specialized fields it as a revelatory as the introduction of complex numbers. Discoveries like this do not happen often.

It turns out the math is very general (so a bunch of fields I have neglected also use schemes). Because it is general it ends up being defined in terms more abstract than polynomials and roots (in terms of morphisms, topologies, Spec, and so on). Schemes are great because they work even over very general concepts (and not because they bring in very general concepts).

(For a text dealing with the algorithmic aspects of ideals and varieties (but, not schemes) I recommend David A. Cox, John Little, Donal O’Shea, “Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra” 3rd Edition, 2008. Schemes are a pretty advanced topic, you can try Robin Hartshorne “Algebraic Geometry”, 1997, but to even use the book you would need a background in commutative algebra.)

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Data Scientist and trainer at Win Vector LLC. One of the authors of Practical Data Science with R.

5 replies

  1. “There is no real number that is a solution to the polynomial x^2-1=0”

    I think you mean the equation x^2 + 1 = 0, which requires complex numbers. x^2 -1 = 0 has {-1, +1} as solutions.

    1. Ooops I did type the mistake x^2-1=0 I meant to type x^2+1=0 throughout the article. Very embarrassing typo, sorry about that. Fixed now, thanks for helping.

      1. Hartshorne is arguably the driest algebraic textbook in wide circulation, even for mathematicians. Mumford’s own Red Book of Varieties and Schemes is specifically made to bridge the gaps in intuition by some analogies to what a scheme is. Commutative algebra is easy to orient in with Miles Reid book.

        Mumford spent his life shaping things to be not only motivated but the motivation intuitive. But when advanced audience may have no intuition of what a complex plane is… even most accommodating sensitive hearts will find it so very hard making things simpler than possible.

        I find attitudes in communities seemingly interested in mathematics more shunning than those of understandably uninterested biologists. Whereas a biologist, as Pachter says, may be quick to look the other way, a garden variety “data scientist” will not refrain from expounding his proclamations about utility of “complex math” beyond complex numbers and honesty of “pencil pushers”. Recently I even seen LeCun most politely but still just-so dismissive of having a rigorous, possible to reason about theory guiding deep learning experimentations (and also politely explaining what Mike Jordan “really meant” when denouncing the hype surrounding big data and deep learning). And then there are ideas ignored in turn by ML “pencil-pushers” themselves where algebraic geometry may abstract graphical models and one may plug all the rich algorithmic and conceptual machinery OP mentioned for selection of statistical models like points on algebraic variety.

      2. Chris, thanks for your book recommendations. Also, I have the deepest respect for Professor Mumford.

        On the other issues in your comment. I’d say curiosity is good in any scientist, data scientist, or mathematician- but it does tend to bring you into fields where you are not expert. I think that means you have to be forgiving of some mis-steps. We probably are more in agreement than you may think in that dismissiveness, arrogance and non-curiosity are not attractive in anyone (be they scientist, data scientist, or mathematician).

        Finally I am not entirely certain why you brought up ‘garden variety “data scientist”.’ In this essay I certainly never brought up data science and I don’t think I so dismissed any other group. So, I have a hard time understanding the portion of vitriol directed back here. Mostly I let your comment through for the other portions, but I found some value in letting the rest of my audience see what I sometimes get in comments, email, and on other forums.

  2. Thanks for this post.

    It indeed brought me further along then the Mumford piece and seemed just about right for what I needed.

    Thanks for the book recommendations.