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Theorems for a price: tomorrow’s semi-rigorous mathematical culture

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Conclusion

Zeilberger has proved some breathtaking theorems [ZB], [Z3], and his W-Z method (joint with Wilf [WZ]) has been a godsend to me [A2] and an inspiration [A3]. However, there is not one scintilla of evidence in his accomplishments to support the coming “... metamorphosis to nonrigorous mathematics.”

Until Zeilberger can provide identities which are (1) discovered by his computer, (2) important to some mathematical work external to pure identity tracking, and (3) too complicated to allow an actual proof using his algorithm, then he has produced exactly no evidence that his Brave New World is on its way.

I regret feeling compelled to write this article. Unfortunately articles on why rigorous mathematics is dead create unintended side effects. We live in an age of rampant “educational reform.” Many proponents of mathematics education reform impugn the importance of proofs, and question whether there are right answers, etc. A wonderfully sane account of these problems has been given by H.-H. Wu [Wu1], [Wu2]. A much more disturbing account “Are proofs in high school geometry obsolete?” concludes Horgan’s article [H]. It is a disservice to mathematics inadvertently to provide unfounded ammunition for the epistemological relativists.

If anyone reading this believes the last paragraph is rubbish because attempts (unknown to me) are currently underway to insert the Continuum Hypothesis or the Theory of Large Cardinals into the NCTM Standards for School Mathematics, please don”t write to tell me about them. I can take only so many shocks to my system.

Finally, wisdom suggests that grand predictions of life in 2193 ought to be treated with scepticism. (“Next Wednesday’s meeting of the Precognition Society has been postponed due to unforeseen circumstances.”) A long-overdue analysis of some of our current prophets has been attempted by Max Dublin [Du]. Especially noteworthy is Dublin’s Chapter 5, “Futurehype in Education. “ I won’t give the plot away, but I recall the words of Claude Rains near the end ofCasablanca: “Round up the usual suspects!”

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  1. Department of Mathematics, Pennsylvania State University, 16802, University Park, PA, USA

    George E. Andrews

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  1. Doron Zeilberger
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Zeilberger, D., Andrews, G.E. Theorems for a price: tomorrow’s semi-rigorous mathematical culture. The Mathematical Intelligencer 16, 11–18 (1994). https://doi.org/10.1007/BF03024696

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