Hebrew numerology, known as gematria, has to be given attention within any work on logic, inasmuch as it is, rightly or wrongly, used by many teachers in Judaism as a method of inference. However, it is rarely a process through which mitzvot or minhagim are legally established (though the interdiction against eating nuts on a certain festival comes to mind), but is rather a neat way to explain certain laws or traditions ex post facto, producing associations between words or concepts which otherwise would remain far apart.
I have not to date looked into the history of ‘gematrial logic’, though I suspect it has gone through a process of development. It is reportedly rooted in the Talmud; but my impression offhand is that it gained prominence in the mystical period following the Zohar, especially in the time of Tverya and Sfat mystics like the Ari haKadosh (R. Yitshak Luria, 16th century CE).
Be that as it may, it is relatively easy for us logicians to devise a way to test such an approach to knowledge, as we shall now show. First, it should be noted that the general justification given for gematrial logic by its theoreticians, which gives it some plausibility, is that the Hebrew language is a Divine creation, antedating the Creation of the rest of the world, since the former was used as the instrument of the latter, as implied by use of the verb Vayomer (and He said) in the first verses of the Torah. Thus, according to this view, the Hebrew language underlies the phenomena of this world in a very deep and significant way, and the (correct) names of things may be conceived as not mere arbitrary appendages of them, but as reflecting their very essence somehow. Since the ordered Hebrew alphabet from early on also served as a list of numbers, and that numbers have arithmetical relations, it may naturally be assumed that the letters have parallel relations.
While not denying outright the truth of such claims, logical science cannot rest content with justifications formulated so vaguely and broadly. Especially tenuous, in the above arguments, is the connection proposed between letter and number; for even if letter sound and shape reflect essences, it does not so easily follow that their corresponding numerical values are also metaphysical facts. In any case, the issue cannot be resolved purely empirically, at least not in a way open for all to see for themselves. It must be resolved by more rational means. Logic proposes a way: consistency checking. We may arrive at an objective evaluation of gematrial logic by systematizing its application and considering whether the results obtained make sense. The first task would be to list the various devices used in gematrial inference, such as those listed below as examples.
a. The letters of the Hebrew alphabet have certain numerical values, a=1, b=2, g=3, etc. The primary numerical value of a word is the sum of the numerical values of the letters composing it.
b. The primary value of a word may optionally, on occasion, be extended by a unit; they say, “plus one for the word as a whole”.
c. The primary value of a word may optionally, on occasion, be shortened by simply adding together the digits of the primary value; e.g. 81=8+1=9.
Clearly, each such device produces a distinct category of numerical value for all words, so that each word has, correspondingly, a number of possible numerical values. Words with the same numerical value are numerologically “equal”, and the things they refer to can be conceptually associated. An obvious general consequence of such a principle is that all words composed of the same letters, in whatever order, are numerologically identical. But more broadly, we can produce a ‘dictionary’ of Hebrew words in which the various numerical values of each word (the primary, the extended, the shortened, etc.) are specified. Furthermore, a corresponding ‘thesaurus’ of gematria can be made which groups together all words having the same numerical value (according to the method of calculation a, b, c, etc. – or irrespective of these methods if it is permitted to equate their results). These ‘books’ would be easy to produce in a modern personal computer with a good spreadsheet program, given the database of Hebrew words.
It is impossible to predict the results of such general calculations and classifications. The results obtained may seem so ridiculously mixed-up as to discredit gematrial logic even in the eyes of its advocates; or some interesting and unexpected regularities may be brought to light capable of impressing even skeptics. My guess is that, for the most part, the words thus grouped together will be so remotely related to each other, whether as synonyms or antonyms or otherwise, that the value of their being so associated will be at best literary; very rarely, if ever, can we expect the words thus grouped together to present an unequivocal conceptual message. Such may even be the ultimate position of Jewish tradition, since it does not categorically grant gematrial logic a general and binding legal status, but mostly uses it for homiletic purposes.
We might also finally mention, in this context, what may be called the ‘new numerology’, which has become fashionable in some circles today. Using computers (though the job could conceivably be done manually), every nth letter (with various values of n) of a Torah text is flagged; e.g. every 10th letter, or every 50th, etc. The letters thus selected are assembled (occasionally, by reversing the order of the letters); and in some cases words or phrases appear, which are not only meaningful in themselves, but which may even bear a relation to the content of the text they occur in or seem to predict later historical events. This is indeed very surprising, and is considered as proving the prophetic origin of the document, since no ordinary human could have intentionally written a document with such properties.
Again, not being involved in such research myself, and not having bothered to study its published results in detail, I cannot pronounce myself concerning it. All I can do is state some of the principles that come to mind, which may or not have been applied:
a. it is not enough to apply the technique to only some parts of the Torah, but it must be done systematically to all of it, and with all possible values of n (the number is limited, since the document is of finite size);
b. an explanation has to be adduced for meaningless letter collections, which presumably occasionally occur; and
c. the technique must be applied to many other documents, too, religious or secular, which serve the function of control samples for the experiment.
Only thus, in perspective, can the results, whatever they are, be evaluated scientifically. In any case, it is to be wondered whether any interpretative dividends from such a technique, devoid of theoretical basis, could be used for Halakhic or even Hagadic purposes. This method is not to my knowledge traditionally counted as a valid hermeneutic technique; so, from the orthodox point of view, it carries no authority. Furthermore, the tool has no predictive value: some letter collections may indeed be meaningful, but they are so only ex post facto (for instance, if researchers had come across the name ‘Hitler’ 100 years ago, they would not have known what to make of it).
The advocates of this new numerology are said to have demonstrated, using statistical methods, that the probabilities of their results arising by chance are infinitesimal. They adduce this in support of the traditional theses: (a) that the Torah is of Divine origin and (b) that it contains all the secrets of the universe and a prediction of all of history. But these inferences constitute illicit generalizations. With regard to the first thesis, we can only at best conclude that passages of the Torah which actually yield impressive predictions are prophetic; passages apparently not yielding such data remain uncertified. With regard to the second thesis, the discovery of a few meaningful character strings does not entail that all science and history are contained in the document; one would have to find all or most known science and history mentioned in it, to dare make such broad claims.
Another comment worth making is: if the Torah were ever proved to be of Divine origin, what would become of faith? Is not uncertainty essential to religion?
 As an acquaintance of mine has suggested, the word perhaps derives from ‘geometry’ (Gk.)
 According to J.E. (apparently). Incidentally, J.E. mentions other exegetic techniques in a similar vein, such as the splitting of words (notarikon) and the transposition or the substitution of letters in words (al tikra… , ela… ; i.e. do not read… , but…). Such changes are not, of course, to be made in the Torah text itself, but allow for variant readings. With regard to splitting words up or transposing their letters, these proceedings are consistent with gematrial assumptions, insofar as the numerical value is unaffected by them. More broadly, such techniques would seem justified to the extent that they are consistent with established etymological and phonemic principles. However, in some cases, the proposed readings seem artificial and fanciful (to me, at least). We should also mention, while on this topic, variations in vocalization, on which R. Akiba, preceded by R. Yehudah b. Roez, relied (yesh em lemikra); but R. Ishmael considered that ‘only the consonantal text is authoritative’ (yesh em lemasoret). Logically, the consonants are empirical data; while the vowels have the status of hypotheses, to be confirmed so far as possible by the entire context.
 See Appendix 5.
 As a system of numbers, this is much less practical than the decimal. The first 10 letters of the alphabet stand for the numbers 1 to 10, the next eight for 20 to 90, and the last four for 100 to 400; thousands may be indicated by an apostrophe. A number like 5755 is written h”nwt’h, reading from right to left 5000+400+300+50+5. There is no letter standing for zero (efes). Arithmetic operations have no sign, but are expressed in words. In my opinion, the lack of zero, the asymmetry of the three numbers 200, 300, 400 (no 500, etc.), and the superfluity of numbers (20 on up), as well as the awkward way available numbers combine to form larger numbers, all point to the antiquity, but also the human origin, of this system.
 Possibly, a three-digit short value might be shortened to two-digits, and then further still to one digit; also, the extended value might similarly be shortened; each such device, if permitted, gives rise to a different category of numerological value. In any case, I know there are still more devices. For instance: letters whose values are ten times greater may occasionally be considered equivalent; this applies to the sets: 1, 10, 100; 2, 20, 200; 3, 30, 300; 4, 40, 400; 5, 50; 6, 60; 7, 70; 8, 80; 9, 90 (this device, incidentally, might be viewed as a corollary of ‘shortening’). Another example: the letters of a word may be spelt out in full, before conversion to numerical values; thus, an a becomes alef, and thereby worth 1+30+80=111; a b becomes bet, and worth 2+400=402; and so forth (this might be called ‘multiplying’, and presumably outcomes of this device can in turn be subjected to other devices).
 I am not sure offhand whether or to what extent different categories of value are comparable: can a primary value be equated to an extended or shortened value? or only to another primary value? In any case, these are technical details we need not go into here, our goal being only to define a methodology of verification.
 Hebrew vowels, composed of dots and dashes, do not (to my knowledge) have numerological values.
 The given name is, I think, ‘Sequences of Equidistant Letters (SEL)’.