Question about the history of mathematics

  • #1


Gold Member
Since this is on the history of philosophy linked to set theory, rather than set theory itself, I presume I can't put it into the Set Theory rubrik.

First, for those who are unfamiliar with the Reflection Principles of Set Theory (or more properly model theory),there are several versions, but they very roughly say that if you have a proper class which satisfies a first-order sentence, then there is some set which satisfies it. That's rough, and lots of caveats need to be added, but this is enough for the following question. (Those of you who are familiar with reflection principles, take the one of your choice, and the following question still holds good.)

Somewhere (many years ago) I seem to remember that one had something similar, albeit not as precise, in medieval European philosophy (much of which would be called theology today) which said that any time that a human is describing a quality of God, then she is really describing some creature of God by a modern logician would call relativising ; even when one tries to cheat and uses terms which would seem to preclude this, such as "the greatest good", one really means "the greatest good that I can think of." So saying that God is infinite really just described the stars, and so forth. (When they were not quite sure what "creature" to attribute a property to, they usually stuck it on some class of angels.)

I don't include a reference because that is what my question is. I can't seem to find a source that says this. If anyone can point me to a source which is freely available online (i.e., a link not leading to a subscription form or requiring payment), I would be grateful.

Answers and Replies

  • #2
What exactly is
I can't seem to find a source that says this.
Pascal, Descartes and Leibniz dealt with god IIRC. A look at Spinoza could be reasonable. too.
  • #3
Thanks for the attempt, fresh_42, but since almost all European medieval philosophers (and even many of the Renaissance ones) wove theology into their works, naming a few of them that did so does not narrow my search much. Also, a medieval philosopher who expounded this version of the reflection principle was not necessarily one that was connected directly to mathematics.
  • #4
Please be more specific.
this version of the reflection principle
Do you mean the idea of god as a collection of man made properties? The atheist among philosophers was certainly Nietzsche, but he doesn't fit the time. Medieval humans had - other than today - a surprisingly specific imagination of religious stuff. This contradicts the abstract description you gave. God was seen as a person.
  • #5
Do you mean the idea of god as a collection of man made properties?
No, quite the contrary. The idea was that the properties of the divinity were out of the range of the perception of mere humans, so that the only thing that men could perceive were the "reflections" of God in lesser things. That is similar to the situation between a set and a proper class; more precisely, between the universe of a model, and the sets. A universe of a model can never be a set in that model (although, of course, it can be a set in another universe.) (If you prefer to keep to theories and sets, then something like an axiom for measurable cardinal versus ZFC.) So in this version, God was not seen as a person; he was more than a person, just as a proper class is not a set. The Greek gods, of course, were a different matter; they were on the level of humans, but the Greeks did not, as far as we know, come up with any sort of reflection principle.
  • #7
Consider beginning your search of theologians with Saint Jerome. Jerome learned Hebrew and studied the Torah and related texts as the basis for Christianity. Spinoza based some of his ideas on translations from Jerome, as did many other scholars including Martin Luther. Luther attempted to reform the established Catholic church of his era and promote the study of history and science with mixed results.

Since your post specified origins of set theory in earlier theology, you should be familiar with Georg Cantor; a devout Lutheran and "founder of the feast", so to speak, of set theory. I would downplay the sensationalism attached to Cantor's later struggles with mental illness, as that no more repudiates set theory any more than John Nash's health problems diminish his insights in game theory and other mathematics.

Returning to medieval scholarship, I strongly recommend finding the best translations of Roger Bacon, considered the leading light among early scientists, cerca 13th Century. Roger Bacon's publications, particularly after 1247 C.E., formed a basis for experimental science and rational thought culminating in the Age of Enlightenment.
  • #8
Thank you for your input, fresh_42 and Klystron, but...
“Per substantiam intelligo id quod in se est et per se concipitur hoc est id cuius conceptus non indiget conceptu alterius rei a quo formari debeat.”
is expressing a concept of self-reference rather than reflection. (Self-reference went down in disgrace, whereas reflection went on to glory.)

I am well familiar with Cantor; he did not come up with a principle of reflection, although he had a couple of near-misses in his struggles with his concept of the "Absolute Infinite", sort of like the near-misses of Guildenstein in Tom Stoppard's "Rosenkrantz and Guildenstern are Dead". In any case, from all that one knows about Cantor, his inspiration are based more on Dedekind and Riemann than on any religious or philosophical source, but since large cardinals were not yet around, the zeitgeist for reflection principles had to wait.

St. Jerome was not medieval. He was of course a reference for many philosophers and theologians, but their number is too great for this to narrow my search. Nor is it practical to sift through all of Bacon's works for the off-chance that it might be alluded to there; I was hoping rather that someone in this Forum might be familiar with the specific instance to which I am referring.

I apologize if I gave the impression that I am researching the origins of set theory in this question; rather, I am searching specifically the appearance of a medieval reflection principle which , although having the tantalizing similarity to the later mathematical reflection principle, almost certainly did not lead directly to it. A case of convergent evolution with a time shift. (I am not trying to connect the two appearances of the ideas: that would be a sort of historical revisionism beloved by many leaders -- for example, as the Shah of Iran claimed a continuous line back to Cyrus the Great.)
  • #9
[Spinoza] is expressing a concept of self-reference rather than reflection.
I don't agree. This here is exactly the reflection principle in my opinion: said:
It follows from these two axioms of Spinoza that if several substances that are different from each other are accepted, something must be based on common features, since the substances cannot be distinguished from one another without a common one. The definition of a single substance can only be done by its differentia from the other substances. However, this would mean that no substance would be comprehensible on its own, but only in relation to the others.

The distinction here, i.e. in your understanding of the reflection principle and self-reference is nitpicking. The reflection principle is a self reference (each definition of the properties of a set is actually a set). The only difference is, that this self-reference is explicitly resolved by semantic means, as classes resolve the paradox of Cantor's sets. Spinoza did the same, except that he failed to emphasize the semantic character of such a resolution.
  • #10
fresh_42 Interesting that you quoted a Latin text without translation, while you felt that it was necessary to translate the German. Actually, it is the other way around: I speak German, but my Latin is rusty. Translations are good for other readers, though.

The quotation from the German wiki
is part of Spinoza's argument (in a proof by contradiction) that
Una substantia non potest produci ab alia substantia
i.e., that one substance cannot be produced from another.

Your implicit argument is interesting: if I understand it correctly, you are saying that if everything is one substance, then a lesser being must share some qualities with the lower being. What is not clear to me (either from your argument or from Spinoza's) is that, if the substance is the same, how you define a "lower being". That is, the idea of reflection incorporates a differentiation, an order (class/set, God/angels, etc.).

I suppose you can stretch a point and see a reflection as a kind of self-reference, although it is stretching it even further to see self-reference as a kind of reflection by eliminating the pertinent clause of the reflection principle to be that the reflected set must be of a lower order than the reflected. That is, the reason the introduction of classes resolved Cantor's paradox was that it eliminated all self-reference. Self-reference and reflection are, to pick a nit, incompatible. The closest one gets to having a kind of reflection (coding) retain a kind of self-reference is in the Diagonal Lemma (the one which allows, IMHO, the niftiest presentation of the proof of Gödel's First Incompleteness Theorem).

But anyway, even if you could somehow infer a sort of reflection principle from Spinoza's writings, the end statement would be a modern deduction. The Latin phrase you cited is not a direct statement of the type of reflection principle that I remember seeing very clearly stated in a form similar to "if one describes God, he is actually describing one of his creatures". This reference is what I am looking for, not a reference from which one can indirectly deduce one. So, although I thank you for the suggestion, Spinoza doesn't fit the bill.

Suggested for: Question about the history of mathematics