250 Bugetin

[17]

[18] [19] [20] [21]

[22]

[23] [24]

[25] 126]

M. Makkaio and G.E. Reyes, First Order Categorical Logic, Lecture Notes in Math. No. 811 (Springer , Berlin-New Yo~ 1977). G.P. Monro, On genedc extensions without the axiom of choice, J. Symbolic Logic 48 (1983) 39-52. C. Negoits, Fuzzy sets in topoi, Fuzzy Sets end Systems 8 (1982) 93-99. A.M. Pitts, Fuzzy sets do not form a topos, Fuzzy Sets and Systems 8 (1982) 101-104. D. Ponasse, Some remarks on the category Fuz(H) of M. Eytan, Fuzzy Sets and Systems 9 (1983) 199-204. G.E. Reyes, From sheaves to logic, in: A. Daigneault, Ed., Studies in Algebraic Logic, Studies in Math. No. 9 ( M ~ , Washington, DC, 1974) 143-204. D.S. Scr~ end R.M. Solovay, Boolean-valued models for set theory, Unpublished notes (circa 1985), D,S. Scott, Boolean models and nonstandard analysis, in: W~J. Luxemburg, Ed., Applications of Model Theory to Algebra, Analysis end Probability (Holt, Rinehart and Winston, New York, 1969) 87-92. L.N. Stout, Topoi and categories of fuzzy sets, Fuzz/Sets and Systems 12 (1984) 169-104. L.A. Zadeh, Fuzzy sets, inform, and Control 8 (1985) 328-353.

Editor's note

Dr. Johnstone has subsequently discovered and informed one of another, I must say excellent, reference; it is:

M. Bert, Fuzzy set theory and topos theory, Caned. Math. Bull. 29 (4) (1986) 501-508.

4. A Rep|y to Peter ,Johnstone's "Open Letter"

I was gratified to see that my article in Fuzzy Sets and Systems had evoked such an emotional response. Your reply points out a num~r of errors which I w(~uld not presume to chaflenge since I am not in any way s professional mathematician. However, the original purposes of my polemic stiff stand. The~ were as follows. First, that the sedes of papers which I surveyed had left something of a mud~Jle and that such muddies usually conceal some ~nterssting core of truth (fuzzy or otherwise). Secondly, that purely engineering considerations had |ed to the intuition that one ought to be able to use fuzzy sets theary (or at least some derivative of it) to formalise some of the 'higher order' processes which knowledge engineers wish to capture. Thirdly, nothing in the papers mentioned, or indeed in your latter, proves that this cannot be dons with s suitable definition of morphisms. Eytan, for example, proposes just such a solution in a so far unpublished paper in which the Kronscker delta is reptaced with an intuitionistic version. Against this it might be argued that the normal logic o f ~ ~ differs from that of toposes (due partly to the semantics of the intermediate truth values)and that a unification of the theories is doomed a priori. However, this is an argument guaranteed to stultify research fully in a~ord with the methods of British empiricism which permeate you¢ letter. My point is that toposes (or aW other mathematical edifice), while they might have a beautiful rigo~us structure designed to give mathematicians years of innocent amusement, find their real justification in their ability to address practical problems. Perhaps the theory (whether it be th , i ~ to~oses or that of fuzzy sets) needs changing in order to accomplish the task.

When you announce a desire to 'get the history straightened out' apart from dismay at having to live in a linear history I feel that you have misunderstood some fundamental facts. Zadeh and the other pioneers of fuzzy sots were not motivated by 'interest in Boolean-valued models of set theory' or any other such e~strsct consideration. The prime motivation was an engineering one expressed succinctly in Zedeh's much quoted principle of incompatibility. Having said this it is of course perfectly correct that there should be a desire to put the intu;~ion on a sound mathematical footing. The search for a well behaved category of fuzzy sets is only one direction that the searc!~ can take. if .'he theory of toposes provides some inspiration to give direction to the search this is not to s~ly that we must accept th~s theory as finishe~t from a practical point of view. Similarly, if fuzzy isets do not form a topos then maybe

Bu//elin 251

it is the theory of fuzzy sets which is at fault. You cannot answer such questions purely by paying attention to mathematical rigour. In particular, i accept that to begin to Perry out this programme one needs to consider the mathemedcai consequences of fuzzifying equality. The question is: how? if we could see how to do this and retain the intuitive appeal of fuzzy sets we would have a reel contribution to human knowledge, if in doing so we made a few errors of rigour then you mathematicians could sort that out over the next hundred years or so, while we engineers goton with applications, if you can lecture me on the history of~fuzzy sate then ~orgive reefer drawing your attention to the history ofthe calculus in this respect,~ Leibniiz and

Newton and their followem hed:a hopelesslyunrigorous notion of,the differential coefficient, it wasn't until D'Alembert c rown later that the notion of limit was saton anything approaching a f i rm foundation, Does this mean that no papers on the calculus should have got past the refereep in those days? -

On the last point of the standards of refereeing in fuzzy set journals I am not really qualified m comment in date~i. Not being an academic I am not disposed to apply purely academic Standards. in the case of my paper | would draw your attention to the fact that the issue it appeared in was a special one consisting of the proceedings of a conference. As such the editors may have taken the view that a desire for rigour should be balanced with the need for the issue to expr~Gs the (;;versity of views expressed and the importance of these views, in fact, there were a number of points of deteii which ! would have corrected with hindsight. The fact that I did rot give this priority was due to the nature of the article as a polemic meant to encourage research, rather then as a tract on pure mathematics - which it was not.

Let me finish by summer|sing what I think the s~ete of play now is. The category theoretical foundations of fuzzy sets theory are in an incomplete and slightly unsatisfactory state, especially from the point view of knowledge engineering applications. Fuzzy sets however are a proven and valuable tool in this respect. The efforts of category theorist~ (especially topos theorists) may well provide a source of inspiration for the resolution of this contradiction, it is incumbent on us engineers to state potential desirable solutions.

i t is incumbent on you mathematicians to produce the ~equired msthematice| structure. Detailed criticism is always of value, mid your comments were useful, but critique alone does not address the ke~ problem.

The failure of recent mathematics to root itself firmly in the practice| world is current0y Pausing intense problems for the mathematical community in terms of funding. That this troubles msthemedcian8 as well as thl~ rest of us is evinced in particular by the current debate on the teaching of mathematics for computer scientists. A particularly worrying example of this came to my attention recently° Cc.'nputer scientists have good reason to be interested in the mathematics of relations in order to be able to discuss the so-ca|led relational date model. This having been the case for some years it is surprising to find no substantial body of work on the category theory of relations. As far as I understand it, mathematics undergraduates are still only told briefly about such things before being rushed on to the lifetime steJdy of functions, deriving as it does from the practical concerns of nineteenth centu~v physics and mathematics. There can tu l l y be no excuse for this 'head in the sand' approach, if I have to mend my ways in order not to be 'laughed at° by an audience of mathematicians, then surely it behoves the entire mathematical community to change its attitude ta avoid a derisory response from the much larger audience which pays the bills.

lan Graham

5. Expert Systems

Expe~ Systems 88, the Eighth Annual Technica~ Conference of the B~itish Computer $ocie~/ Specialist Group on Expert Systems will be held at the Metropole Hotel in Brighton from 13th-15th December, 1988. It will be preceded by one ~ d ~No day tutoria|s in the same location on Monday and Tuesday, 12th end 13th December 19f~8. The conference wil| focus