Euclids Elements

Euclid's Elements epitomise an enormously important insight of Greek philosophy: firstly, you must agree on the definitions of the key terms you are going to discuss, otherwise, your discussion will just go around in circles. Secondly, you must agree on the logical principles which bind those definitions together [the axioms], otherwise, again, your discussion will go around in circles.

Once you have agreed the definitions of the key terms, and the logical rules by which they can be manipulated, you can go on to elaborate theorems (ie knowledge).

This is true for ALL branches of science and mathematics, and just as much today as it was two and a half millennia ago.

With regard to mathematics, there are three axiom sets you need to be familiar with (and unfortunately, few mathematicians are really familiar with any of them).

1. The Peano/Dedekind axioms of arithmetic.

2. The axioms of mathematical logic.

3. The axioms of set theory.

By the way, Euclid's Elements orginally included no diagrams. Its arguments were entirely based on logic.

Reply to alan1000 thanks a bunch, however shouldn't some of the axioms by Euclid be included in the list since arithmetic is grounded in geometric proofs? Not that I have much of a right to formalize a question.

Quoting alan1000

(and unfortunately, few mathematicians are really familiar with any of them).

Is this opinion or fact? Please state your references if the latter.

Reply to jgill Is that the question you want to be asking?

I'm curious. As a retired prof I don't think it's accurate.

(and unfortunately, few mathematicians are really familiar with any of them). — alan1000


"Is this opinion or fact? Please state your references if the latter. "

My apologies! I was too facetious.

Quoting Thinking

So just took a free online class at Hillsdale college about logic and reasoning and how learning geometry from Euclid and his elements help with that. My question is how in what ways do you think Euclid's elements helped you out the most? I can tell you it helped me learn geometry a bunch but not so much on the logic/reasoning side. What say you pilgrim?

Same here! I enjoyed math in my teens, especially geometry and I, sadly, failed to make the connection between geometry in particular, mathematics in general and logic. My life would surely have been greatly enriched had I realized that.

For my money, Euclid's place in logic is the same as (say) Arnold Schwarzenneger's place in bodybuilding - both were paragons in their respective fields. That being so, they become, by virtue of their excellence in their areas of expertise, perfect reference points for beginners to get a feel of what it's all about. @alan1000 is spot on.

Also relevant seems to be the relationship between mathematics and logic? Are they one and the same thing? After all, the content of the branches of mathematics (theorems) seem to be arbitrary - mathematicians seem at liberty to begin with any axiom of their choice - and the only permanent fixture in the mathematical universe is logic. So, the natural question is, is math just logic in its [Euclid's] elements?

Reply to TheMadFool I thought I foresaw that axiom loophole which could be overused and abused. I am currently working on perfecting the skill of logic and reasoning just how Euclid, Aristotle, and my good pal Abe Lincoln had done.

Quoting TheMadFool

I, sadly, failed to make the connection between geometry in particular, mathematics in general and logic

I would blame the educational (which is actually social/political) system for that common insufficiency that you and I both share.

So, the natural question is, is math just logic in its [Euclid's] elements?

I feel like if math is a universal language then it should express properties that the universe contains. The universe I think isn't simply just cold logic but also contains feelings, reason, and everything in between. So, more accurately I would like to say mathematics should be expressed through rationality rather than just simply logic in itself. On that line of thinking everything outside of what we couldn't use with mathematics would be irrational, or a term I like to use is anti-rationality.

I am curious as to what publisher's book you used as your copy of Euclid's Elements. I'm curious because the first edition of Euclid's Elements that I read was the Barnes And Noble edition. In that edition, Book 1 runs from page 1 to page 217, and my experience was the same as yours. I didn't see the logic that I expected to see. That edition is full of commentary. I'm sure that commentary is extremely useful for some readers. To me, it was extraneous information.

After that, I read the Green Lion Press version of the Elements. Book 1 runs from page 1 to page 36. It's mostly a bare bones translation of Euclid's definitions, common notions and propositions - I say mostly because the book contains bracketed references to previous propositions, common notions, etc. as justifications for the arguments used in the proofs of the propositions. What I saw in that book was an elegant logical construction of a fairly complex argument, the proof of the Pythagorean Theorem and its converse.

The logic of Euclid's argument came through much more clearly in the uncluttered book.

Quoting Thinking

I feel like if math is a universal language then it should express properties that the universe contains.

[quote=Galileo Galilei]Mathematics is the alphabet with which God has written the universe[/quote]

Mathematical Universe Hypothesis (Max Erik Tegmark)

Quoting Thinking

The universe I think isn't simply just cold logic but also contains feelings, reason, and everything in between.

I know where you're coming from. I think I have a name for it - Spock Paradox which is the hard-to-explain Spockmania given Spock is a personification of cold logic. Think of it, if logic were really that emotionally empty would people actually spend their whole lives studying it? Plus, what if there are no good reasons to be emotional about anything? Yet, philosophy means love of wisdom! I guess, as the ones who seem to know what they're talking about say, it's complicated.

Quoting Thinking

So, more accurately I would like to say mathematics should be expressed through rationality rather than just simply logic in itself.

Explain the difference between rationality and logic

and

what the heck is anti-rationality? Irrationalism?

Reply to TheMadFool logic is very systematic and straightforward with well defined rules that you can follow. Reason on the other hand gets more complicated in that there are less solid guidelines to follow in using it. Rationality, however, contains the whole complex of human energy that you work with and more importantly the feelings that we all have. Through rationality I use logic and reason but also feelings to determine the truthfulness of my conclusions.

If I end with a philosophical conclusion feeling helpless, sad, negative, depressed, etc... I know by trusting my feelings that the conclusion I came up with is false and shouldn't be implemented. If instead I get a conclusion that gives me joy and an inspiration to co-create something new into my life, then I came up with something good that will help people and the world. The goal of all my philosophical conclusions is to confirm the perfection and beauty that is life!

Now anti-rationality is when you leave out those important feelings and come up with spockmania as you may call it. By following this example you get the artificial reality we have created today which destroys and exploits the natural world (gods thoughts) in order to sustain itself for a temporary period of time until it can't no longer.

On the other hand feeding those [I]rational[/I] conclusions with your thought and actions makes you nourish and cultivate the natural world that we are an important part of. For the natural world is eternal by way of auto-genesis and provides objects that were created by a vastly superior intelligence than the artificial ones that are created today and, consequently, by interacting with the natural world we can attain higher truths that will help reciprocally serve to create the paradise on Earth(or eden if you will) that is the birthright for every given man. This is our journey back to our pristine origins. Back to knowing god through his creations.