fibonacci sequence nature split end peeling

Well, at least the way I thought about it for Mathematica was this: I tried to think of all the possible computations that people might want to do, and then I tried to see what chunks of computational work came up over and over again.

The good ones tend to get invented pretty much all at once, normally by just one person. But computer languages have historically been different. Normally, languages arise through some sort of gradual historical process of consensus. And basically the way that people seem to communicate anything sophisticated that they want to communicate is by using some kind of language. One of the big achievements of Mathematica, as probably most of you know, was to figure out a very general way to have the computations inside work and also be practical-based on just doing transformations on symbolic expressions, with the symbolic expressions representing data or programs or graphics or documents or formulas or whatever.īut just being able to do computations isn’t enough one also has to have a way for people to tell Mathematica what computations they want done. There are really two parts to that: how the computation inside works, and how people get to direct that computation to do what they want. And I guess in general my goal with Mathematica was somehow to take the general power of computation and harness it for all kinds of technical and mathematical work. Because with Mathematica, one of my big goals has been to take another big step in what one can think of as the systematization of mathematics. But after that, pretty much nothing.īut in a sense it’s not surprising that I’ve been interested in this stuff. And at the end of the 1800s and the beginning of the 1900s, when abstract algebra and mathematical logic were really getting going, there was another burst of interest and activity. Babbage wrote one of his rather ponderous polemics on the subject in 1821. So Leibniz and other people were interested in these things in the mid-1600s. In the past, the times these things have ever been thought about, even a bit, have mostly coincided with various efforts to systematize mathematics. Because it turns out that so far as we could all tell, there’s almost nobody who’s alive today who’s really thought that much about basic issues regarding mathematical notation. Cajori-he has been dead for no less than seventy years.Īnd I think there weren’t too many other possible choices. But upon investigation, it turned out that there was a logistical problem in inviting the esteemed Dr. And there was an obvious candidate speaker-a certain Florian Cajori-author of a classic book entitled A History of Mathematical Notation. When this conference was first being put together, people thought it would be good to have someone talk about general issues of mathematical notation. What I have discovered about what generalizations of mathematics might be like. Mathematical notation may have limited the scope of mathematics, and some of I will discuss the extent to which traditional The scope of a particular human language is often claimed to limit the scope Might make use of more general structures, and whether human cognitive abilities Ordinary language involves strings of text mathematical notation often also With these in APL, Mathematica, theorem-proving programs and other systems. I will discuss attempts to develop notation for algorithms, and experiences Traditional mathematical notation represents mathematical objects but not mathematical Handling such expressions and making them easier for humans to understand.

Large mathematical expressions-unlike pieces of ordinary text-are often generatedĪutomatically as results of computations. Of mathematical notation that we discovered in doing it.

I willĭiscuss some of the key ideas that made this possible, as well as some features Understanding something very close to standard mathematical notation. Natural language-would be extremely difficult for computers to understand.īut over the past five years we have developed in Mathematica capabilities for Given its historical basis, it might have been that mathematical notation-like To mathematical notation, while others do not.

That have been discovered for ordinary human language and its history apply I will discuss the extent to which mathematical notation is like ordinary human Peano and others, to its widespread use in the nineteenth and twentieth centuries. I will review how it developed, with precursors in antiquityĪnd the Middle Ages, through its definition at the hands of Leibniz, Euler, Most mathematical notation now in use is between one andįive hundred years old. Transcript of a keynote address presented at MathML and Math on the Web: MathML International Conference 2000

Fibonacci sequence nature split end peeling