A new theory of Parallels

[ermete22]

In the A new theory of Parallels Carroll writes: “This field of mathematical research, with all its wealth of hidden treasure, is all too apt to yield nothing to our research; for it is haunted by certain ignes fatui—delusive phantoms that float before us and seem so fair and are all but in our grasp. . . . Alas for him who has been turned aside by one of these spectres, who has found a music in its mocking laughter and who wastes his life and energy in the desperate chase.” Which turns out particularly interesting as in this treaties he is trying to prove some conjecture of the classical geometry (what we should call an axiom) of course without success. The axioms of classical geometry are the “delusive phantoms” he is talking about, propositions taken as true a-priori to develop classical geometry, and which have no justifications. The result of his opinion was necessarily a sense of relativism of geometrical truths based on axioms which cannot be proved.
Carlo

[johntufail]

Hi Carlo!

How could I have missed this quote!

I mentioned in an earlier mail my theory that at an early stage in his career, Carroll began to move away from the study of maths in and of itself, to the study of existential meaning in math. How can one, create a formal language that can validate a premise or axiom. For as you point out here, it is eminently possible to build a whole theoretical universe on the basis of a completely false premise.

The example, I gave, of course was the premise that each individual behaves in an economically rational manner as the basis of modern liberal economic theory.

I think Carroll, however, was most concerned about the consequences of axioms about the natue of God. He explores these at great length, both in his books and his lettes, most notably to Mary Brown.

JT

[ermete22]

Hi John,
I am trying to make some order in the various topics as the subjects tended to mix. It was for sure my responsibility as I had just some experience in moderating strictly scientific forum; which is much simpler.
About math, I listed two completely different works by Carroll, the one about Determinants and this Treatise because they represent two completely different approaches to mathematics.
An Elementary treatise on determinants (ETD) is applied mathematics, which was certainly the starting point for Carroll, for a number of reasons: this is the kind of mathematics you are normally asked to study unless you are a professional mathematician. It is evident, by looking at math works by Carroll, that is was excellent in this kind of math (also the comments of the directors of the schools he attended confirm this hypothesis.) Solving and inventing mathematical problems is one of Carroll’s talent.
But he moves to Oxford and not to Cambridge, where there existed the best British mathematician of his epoch. Why?
The first answer comes to one’s mind, is of course because of his father’s will and the presence of Pusey at the College previously attended by Carroll’s father.
By moving to Christ Church College, Carroll simply exactly reproduces the choice made years before by his father.
The other talent the young Carroll was for poetry, writing, drawing.
One can conclude without seriously risking of being wrong that Christ Church College was the potential burial of all his spontaneous talents. Nobody has never posed the following simple question: what kind of mathematician would have been Carroll if he had moved to the right place, that is Cambridge? The state of math in Oxford was really bad, conservative and irrelevant. One must consider that, when he moved there, he was very young and surely needed to be directed in some sense by some competent mathematician, who did not exist in Oxford to my knowledge. Nobody attracted his attention to the new trends of mathematics. In some sense he was left alone, with old mathematical theories; all of us know how important is in the life of a young person with some potential talent, to meet someone who attracts his/her attention towards the right research development. This did not happen to young Carroll.
Counterfactuals are normally criticised, but I will try to avoid them and reduce my conclusions to the fact that, while before entering Christ Church College, all teachers reports confirm an excellent mathematical talent of the young Carroll, the choice of that very college was paradigmatically wrong. And add another question: is the fixation of Carroll on the fundaments of old geometry is somehow due to the impossibility of finding an outlet of his mathematical talent towards innovative, advanced mathematics?
Carlo

[ermete22]

Hi John,
I am trying to make some order in the various topics as the subjects tended to mix. It was for sure my responsibility as I had just some experience in moderating strictly scientific forum; which is much simpler.
About math, I listed two completely different works by Carroll, the one about Determinants and this Treatise because they represent two completely different approaches to mathematics.
An Elementary treatise on determinants (ETD) is applied mathematics, which was certainly the starting point for Carroll, for a number of reasons: this is the kind of mathematics you are normally asked to study unless you are a professional mathematician. It is evident, by looking at math works by Carroll, that is was excellent in this kind of math (also the comments of the directors of the schools he attended confirm this hypothesis.) Solving and inventing mathematical problems is one of Carroll’s talent.
But he moves to Oxford and not to Cambridge, where there existed the best British mathematician of his epoch. Why?
The first answer comes to one’s mind, is of course because of his father’s will and the presence of Pusey at the College previously attended by Carroll’s father.
By moving to Christ Church College, Carroll simply exactly reproduces the choice made years before by his father.
The other talent the young Carroll was for poetry, writing, drawing.
One can conclude without seriously risking of being wrong that Christ Church College was the potential burial of all his spontaneous talents. Nobody has never posed the following simple question: what kind of mathematician would have been Carroll if he had moved to the right place, that is Cambridge? The state of math in Oxford was really bad, conservative and irrelevant. One must consider that, when he moved there, he was very young and surely needed to be directed in some sense by some competent mathematician, who did not exist in Oxford to my knowledge. Nobody attracted his attention to the new trends of mathematics. In some sense he was left alone, with old mathematical theories; all of us know how important is in the life of a young person with some potential talent, to meet someone who attracts his/her attention towards the right research development. This did not happen to young Carroll.
Counterfactuals are normally criticised, but I will try to avoid them and reduce my conclusions to the fact that, while before entering Christ Church College, all teachers reports confirm an excellent mathematical talent of the young Carroll, the choice of that very college was paradigmatically wrong. And add another question: is the fixation of Carroll on the fundaments of old geometry is somehow due to the impossibility of finding an outlet of his mathematical talent towards innovative, advanced mathematics?
Carlo

[ami]

Hi Carlo,
very good idea to open separate posts for Carroll's theory of parallels and the book on determinants.
It is true that things are very complex when it comes to Lewis Carroll.
Your post for instance deals mainly with mathematics in Cambridge and Oxford, and why did Lewis Carroll go to Oxford instead of Cambridge. Maybe we have to put this in a separate post too on "Mathematics in Cambridge and Oxford" and leave this one just on Carroll's theory of parallels.
My opinion on the subject is just as you said, that Carroll went to Christ Church just like his father.
True Cambridge was better in mathematics (maybe thanks to Newton's influence) but we don't know whether Carroll (and his father), despite his early abilities, was necessarily interested by mathematics alone. He was also good in Classics and Oxford was better in this field.

[johntufail]

Hi all, Just to say that I have posted a response to these mails on the Philosophy site.
JT

[ermete22]

Hi (I posted the same message somewhere else in the forum)
When talking about Carroll’s geometry works, one must never forget his fixed interest was in Axioms. In A new theory of Parallels, which I read many years ago, he tried to find a demonstration of a classical Euclid’s conjecture, starting from what he considered a self-evident truth, but which unfortunately was not such.
To understand Carroll’s story one must realise that he believed in the identity

AXIOMS=SELF-EVIDENT TRUTHS

For him the problem was the identification of such self-evident truths. This is his major error in all the development of his logical (and geometrical) efforts. In modern logics (and geometry) the problem id that building an adequate formal system (Axioms included) and then find out interpretations of the formal system. You change, as an example, an axiom, look at the logical consequences and possibly observe that the new system captures some features of a piece of reality.
Carlo