Infinite Math

 Mathematicians have expanded category theory into infinite dimensions, enabling new connections

among sophisticated mathematical concepts.

This stopped me in my tracks. I had seen this problem before. In fact, the chal­lenge is more than two millennia old, at­tributed to Plato by way of Plutarch. Astraightedge can be used to extend a line segment in any direction, and a compass can be used to draw a circle with any radi­us from the chosen center. The catch forthis particular puzzle is that any points or lengths appearing in the final  rawing must have been either present at the start or constructable from previously provid­ ed information. To double a cube’s volume, you start with its side length. Here that value might as well be 1 because it is the only unit of measurement given. To construct the larger cube, you have to figure out a way to draw one of its sides with the new re­­quir­ed length, which is 3 √ 2 (the cube rootof two), using just the straightedge and compass as tools.It is a tough problem. For more than 2,000 years no one managed to solve it. Finally, in 1837, ierre Laurent Wantzel explained why no one had succeeded by proving that it was impossible. His proof used cutting-edge mathematics of the time, the foundations of which were laid by his French  ontemporary Évariste Galois, who died at 20 in a duel that may have involved an unhappy love affair. At the ripe old age of 20 myself, I had achieved considerably less impressive mathemati­ cal accomplishments, but I at least under­ stood Wantzel’s proof. Here is the idea: Given a point as the or­ igin and a length of distance 1, it is relative­ ly straightforward to use the straightedge and compass to construct all points on a number line whose coordinates are ratio­ nal numbers (ignoring, as mathematicians tend to do, the impossibility of actually plotting infinitely many points in only a fi­nite amount of time).

Wantzel showed that if one uses only these tools, each newly constructed point must be a solution to a quadratic polyno­ mial equation  a x 2 +  b x  +  c  = 0 whose co­efficients a , b a nd c  a re among the previ­ously constructed points. In contrast, the point 3 √ 2 is a solution to the cubic poly­nomial x 3 − 2 = 0, and Galois’s theory of “field extensions” proves decisively that you can never get the solution to an irreducible cubic polynomial by solving quadratic equations, essentially because no ower of 2 evenly divides the number 3.Armed with these facts, I could not re­ strain myself from engaging with the manon the street. Predictably, my attempt to explain how I knew his problem could not be solved did not really go anywhere. In­ stead he claimed that my education had left me closed-minded and unable to “think outside the box.” Eventually my girlfriend managed to extricate me from the argu­ment, and we continued on our way. But an interesting question remains:How was I, a still-wet-behind-the-ears un­dergraduate in my third year of university study, able to learn to comfortably manip­ulate abstract number systems such as Ga­lois’s fields in just a few short weeks? This material came at the end of a course filled with symmetry groups, polynomial rings Step 4: Use a straightedge to draw a line through point 2 on the horizontal axis and point 3 on the vertical axis. Use compass to make two perpen­ dicular lines to construct a parallel line between the vertical point 1 and the position of 2/3 on the horizontal axis.

PYTHAGOREAN THEOREM 

and related treasures that would have blown the minds of mathematical giants

such as Isaac Newton, Gottfried Leibniz,Leonhard Euler and Carl Friedrich Gauss.How is it that mathematicians can quickly teach every new generation of undergrad­ uates discoveries that astonished the pre­vious generation’s experts?Part of the answer has to do with recent developments in mathematics that provide a “birds-eye view” of the field through ever increasing levels of abstraction. Category heory is a branch of mathematics that ex­plains how distinct mathematical objects can be considered “the same.” Its funda­mental theorem tells us that any mathe­matical object, no matter how complex, isentirely determined by its relationships to similar objects. Through category theory,we teach young mathematicians the latest ideas by using general rules that apply broadly to categories across mathematics rather than drilling down to individual  laws that apply only in a single area.As mathematics continues to evolve,mathematicians’ sense of when two things are “the same” has expanded. In the past few decades many other researchers and I have been working on an extension of category theory to make sense of this new expanded notion of uniqueness. These new categories, called infinity categories ( ∞ -categories), broaden category theory to infinite dimensions. The language of ∞ -categories gives mathematicians power­  ful tools to study problems in which rela­tions between objects are too nuanced to be defined in traditional categories. The per­ spective of “zooming out to infinity” ffers a novel way to think about old concepts and a path toward the discovery of new ones.

As mathematics continues to evolve,mathematicians’sense of when two things are“the same” has expanded.