# What mathematical ideas/concepts became obsolete due to technological progress?

As technology evolved, some ideas and methods became obsolete. What mathematical ideas entered this state due to technology progress?

We could consider that doing some mathematical operations done by hand are obsolete because we can do them on computer, for example. But these operations still exist, they're only being processed through computers now.

Is there something on mathematics that entered a complete state of abolishment due to technological progress?

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Log tables might be an example, if you count it as a "mathematical idea". – Alex Becker Sep 8 '12 at 3:28
@Alex: so would slide rules, but in both cases the mathematical idea is that of the logarithm, which has certainly not become obsolete. – Trevor Wilson Sep 8 '12 at 3:41
@TrevorWilson Yep. In this case, I guess it's only a change of interface. – Voyska Sep 8 '12 at 3:43
Along log tables, abacus and quipo could be mentioned. – Sniper Clown Sep 8 '12 at 6:23
You might take a look at a 19th or 18th-century arithmetic textbook, and see what topics are treated there that have fallen by the wayside since. There are many such texts available in Google books. – MJD Sep 15 '12 at 19:32

Mathematical concepts are never abolished by technology. Say like this: you cannot precisely calculate infinite series (and test convergence) by technology, but by logic. Technology is developed relying on formal means: you can make a language parser for math objects (e.g. functions) and apply processes like them (say: integration and derivatives, equation systems resolution). So technology is bound to formal means. However, technology formulates needs to change the way we calculate everything:

• Simplex-Dantzig is a method for resolving equations systems with bounded variables, and finding which one is the max or min value for the overall system or target. But equations, maximum, minimum, are not dead concepts or ideas.
• Square Root has changed a lot of times to use iterative methods. But it is still the square root. Furthermore: the math idea is always more precise in result than the way technology can calculate it (tecnology will never be able to resolve irrational numbers since you need infinite memory).
• The way we iterate to calculate $P(x)$ where $P$ is an arbitrary polinomial is different in computers, changing from an analytical process to a sequential process:

Say $P(x) = 3*x^4 + 4*x^2 + 5*x + 1$

Analytical solution for $P(3)$: $3*3*3*3*3 + 4*3*3 + 5*3 + 1$

Technological solution:

# this is valid python code
P = [1, 5, 4, 0, 3]
x = 3
power = 1
result = 0
for each a in P:
result = result + power * a
power = power * x
#result has the value


But they are still polinomials!

So, methods will vary a lot. Some logarithm implementations in computers use Taylor Series instead of log tables.

However they have no power at all to deprecate concepts, since a concept is not related anyhow to the way it is calculated.

Concepts may die for other, non-techie, reasons:

• Being superseeded by other concepts, usually having better scope (e.g. addition in complex numbers is not the same as scalar addition, because the domain is different, although the name is the same), or resisting paradoxes (it happened with sets, and also with integration regarding Dirac's Delta).
• Being isomorphic concepts.
• Their author and all the people knowing that concept, die.

However, if you take concept only by its name (i.e. sets, integration, addition, function) then they don't die, but change their content. And certainly: names don't die by the in-use technology.

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If you open an applied statistics (or experimental design) book written around 1940-1950 you will find a lot of hand-calculation formulas which nowadays can be safely forgotten!

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It seems calculation with Roman numbers is an obvious example. Or is it too early?

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Nobody ever seriously calculated with Roman numerals. The Romans used abacuses for calculation, which is why it's called "calculation". – MJD Sep 15 '12 at 22:58

While I wouldn't call it obsolete, a lot of the theory of formal languages, parsing and syntactic analysis has lost much of its interest and most of its practical value because the practical problem of parsing is solved sufficiently well.

The theory was originally developed because people truly didn't understand what sorts of languages could be parsed, or how to do it efficiently. But by the beginning of this century, a programmer faced with the task of designing the input syntax for a program can just borrow any of several prefabricated designs, for example something XML-like or something C-like, and then use a prefabricated parser, or, if worst comes to worst, a prefabricated generator for parsers.

These in turn have been enabled by technological advances in computer hardware and programming languages that eliminate the need to sharply bound memory usage. Programmers used to expend a lot of effort designing languages that could be parsed without recursion. As far as I know, nobody cares about this any more.

Parsing has been called one of the great success stories of CS: the theory was developed, the problems were solved, the solutions were implemented, and they are now readily available to programmers, whether or not the programmers know the theory. So the need to know the theory is now close to zero.

Similarly, the actual theory of regular languages has been to a large extent obsoleted by the widespread adoption of canned pattern matching libraries and canned syntactic analysis libraries, and by programming languages that have these libraries built in.

A couple of anecdotes in support of this:

1. I attended a talk a few years ago by John Hopcroft, author of one of the standard undergraduate texts on formal languages and automata theory, in which he related finding himself in the ironic position of trying to convince the Cornell computer science department to eliminate the requirement that undergraduate CS majors take an automata theory course. He said it just wasn't important any more, and certainly not as important as many other things they could be studying.

2. The frequency of questions on this web site about formal languages and automata theory has increased sharply in the past few weeks, as a new crop of CS undergraduates has started getting their first homework assignments. You see relatively few questions on these topics during the summer because nobody but CS undergraduates really cares about these topics.

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I bought Stephen Wolfram's NKS - By reading it, I thought automata was one of the most important things of the century. – Voyska Sep 15 '12 at 19:51
@GustavoBandeira There are a lot of answers to that, but the simplest one is that Wolfram is talking about a completely different kind of automaton that has nothing to do with the theory I am discussing here. – MJD Sep 15 '12 at 19:54
Oh, ok. I thought it was related somehow. – Voyska Sep 15 '12 at 19:57

I'm not sure if this answers your question, but the digit-by-digit decimal calculation of square roots taught to me when I was in elementary school is no longer taught because of the ubiquity of electronic calculators. (If I'm not mistaken, this procedure was first discovered by Thomas Harriot and published in his 1631 book Artis Analyticae Praxis.) Most, if not all, present-day computers use a different algorithm to calculate square roots.

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+1. More generally, technological progress sometimes leads to a change in algorithms used. – Mechanical snail Sep 8 '12 at 8:07
"Most, if not all, present-day computers use a different algorithm to calculate square roots." Yes, but it's not the algorithm you linked, because that's a (rather inaccurate) approximation. Most present-day FPU's have a sqrt instruction (for example SSE's sqrtss), which is most likely (depending on processor vendor) a digit-by-digit algorithm hardcoded in circuitry, plus some IEEE exception handling. – orlp Sep 8 '12 at 10:12
Although actually using the algorithm is obsolete, I think the algorithm could still be used as a motivation for a more efficient solution (say, binary search). I don't know if binary search is the method of choice for performing the computation in practice (probably not), but it surely is used in more complicated situations. On a more philosophical note: suboptimal and impractical ways of doing computation rarely become truly obsolete, since they are frequently a basis of more subtle methods, which often can't be understood well without the initial intuition. – Jakub Konieczny Sep 8 '12 at 10:18
@nightcracker, it seems you are correct. I did not notice that the link I provided was for an algorithm implemented in software, and not in hardware. I'm removing the link. Thanks for the clarification. – Joel Reyes Noche Sep 8 '12 at 13:17

Perhaps numerical methods programed on a computer has diminished the motivation to find exact solutions to particular problems such as integrals, differential equations, and algebraic equations.

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At the risk of irking the older generation, most of algorithmic mathematics dedicated to punch-card computing is now obsolete, at least in the subtleties directly related to punch card machines. Also, the price that the journal in the above link wishes to charge for that article is very obsolete.

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I don't think it's possible for a piece of mathematics to become obsolete. As an object of mathematical research, a piece of mathematics can be superseded only by another piece of mathematics, in which case we would say that it has been improved, extended, or generalized, and not that it has been made obsolete.

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I don't think computer can abolish the significance of actual mathematics. If ever the computer "abolish" or change something, it will likely make obsolete not the math but the practice of mathematics. – William Sep 8 '12 at 3:59
@William: I agree. I would say that a piece of technology (e.g., computers) could make an application of math obsolete, but math by definition has interest outside of its applications. – Trevor Wilson Sep 8 '12 at 4:03
If computers make the practice of math obsolete, we are going to be obsolete too. – Voyska Sep 8 '12 at 4:25
"Nine times seven, thought Shuman with deep satisfaction, is sixty-three, and I don't need a computer to tell me so. The computer is in my own head. And it was amazing the feeling of power that gave him." – Dan Neely Sep 8 '12 at 16:46

I am of the belief that computers have played a very minor role in modern mathematics compared to it influence in science, social science, technology, and culture.

Fundamentally, the benefits of computers is speed, efficiency, and accuracy in computation and calculation. Computers can do a millions of computation in the time it take us to do a single one. (Just imagine how much computation is needed to load this webpage.) You can imagine how much science and technology have advance since people can now do things and obtain knowledge millions of times faster.

The above describes how computers have been useful in many areas. It is because people knew how to solve the problem and realized the problem consists of absurdly large amount of individually simple tasks. Hence a computer can be used to quickly and accurate preform these computation, and obtain results in an instant rather than years.

In mathematics, there are known examples of theorems proven using computers to analyze thousands of cases. This is because the theorem has been reduced to many relatively simple and computational tasks. However, I believe that for most of the important research in abstract mathematics the difficultly lies not in the prodigious amount of sheer computation, but rather an understanding of the problem. Mathematician do not yet understand the problem to be able to divide the question in many, many individually simple tasks for a computer to calculate. Moreover, many open questions in mathematics are more conceptual and do not fit the computational nature suitable for computer calculations.

There are probably areas of mathematics that have greatly benefitted and have been revolutionized by computers; however, there is no ubiquitious presence of computers in mathematics as seen in many of the other ares of science and social science.

The above discussed the computers mostly used as a computation aid. There are some work in using computers to generate or assist in the proof of theorems. But they have not been used to the extent that they have "abolished" the traditional mathematicians proof.

Somewhat relevant to computers, mathematicians have recently begun to study computability itself and the computable aspect of areas of mathematics. Some of this likely preceded the computers. These would include the study of algorithms, Turing Machines, Oracles Machines, computable functions, the Turing Degrees, complexity theory, effective randomness, computable model theory, etc.

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