Even implicitly? My recollection is that in the Euclidean approach lines could not be added to squares because different geometric objects can't be added. Which makes sense when the conceptual reference frame is geometry. When instead an algebraic approach is adopted x squared can be added to x to create an equation. I am asking because dimension seems to play a role in metabolic scaling. It might also play a role in the expansion of cosmological space. I am curious about how Euclid treated the issue of dimension.
Euclid talked about point, line, plane, and space. There is no doubt the concept of dimension was known. On the other hand, contrary to what seems naturally today, not every dimension was considered to pertain to the realm of mathematics. Actually, there have been a huge gap of two thousand years between the old Greeks and modern mathematics. Earlier, mathematics were somewhere between natural science and philosophy. They were an idealisation of the real world, and couldn't stray very far. Even Leibnitz, recent enough for us, was looking for a proof that a space can't have more than three dimensions. After Bool algebra and formal logics, mathematics changed drastically their paradigm. Cantor set theory was based on the principle that a set can contain anything, including itself, which wasn't the usual conception before, that is, a set is a collection of objects that have a characteristic in common. Now we can consider the set RxRxRx…, and make a mathematical reasoning on what we get, without any limit and in an absolutely abstract way.
Algebraic geometry has been introduced by Descartes not so long ago. At the time of Euclid, nothing could be added anyway. Algebra itself was invented by the Arabs, in the middle of the last millennium. The old Greeks knew the surface and the volume of some figures, however they didn't linked them to calculus, but to physical concepts like weight.
Thank you to Claude Pierre Masse (is there a way to add the accent over e on a RG reply?)
I should have read Euclid Book X more closely. Reference here is to the Heath edition of The Elements. A handy table of contents to The Element can be found at:
https://mathcs.clarku.edu/~djoyce/java/elements/toc.html
Book X begins with definitions. In Heath, volume 3, page 10. Here, copied from the website above:
Definition 1.
Those magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable which cannot have any common measure.
Definition 2.
Straight lines are commensurable in square when the squares on them are measured by the same area, and incommensurable in square when the squares on them cannot possibly have any area as a common measure.
Definition 3.
With these hypotheses, it is proved that there exist straight lines infinite in multitude which are commensurable and incommensurable respectively, some in length only, and others in square also, with an assigned straight line. Let then the assigned straight line be called rational, and those straight lines which are commensurable with it, whether in length and in square, or in square only, rational, but those that are incommensurable with it irrational.
From the definitions one can infer that Euclid assumes that areas and lines are incommensurable which is kind of saying that one dimensional objects cannot be added to two dimensional objects. If I am right to look at it like that. Another way of putting it is that a linear combination of geometric objects with areas leads to geometric object that is the sum of those areas. One cannot in that manner form a linear combination of an area and a line.
I am not sure this characterization is correct; this is a tentative and preliminary attempt.
As I understand it, the squares are the ones an edge of which is the considered straight line. Then the areas of the squares associated to two different straight lines are compared. It is indeed not the same as comparing the lengths of the lines, since two straight lines with a length ratio of √2 are inconmensurable in length, but are commensurable in square.
é is Alt 0233
Dear Robert,
Claude Pierre has it exactly right.
The development of our collectively intelligible mathematical language was progressive, really starting with the first idealized geometric shapes that the Greek abstracted from generalizing the less than perfect shapes and volumes that they observed in our environment.
From this idealized first step, the first mathematical equations began to be elaborated and the language progressively evolved to the state it is in today.
You will find a description of its possible evolution from the first idealized representations generated by the use of articulated languages in this recently published paper, titled "The Mechanics of Conceptual Thinking":
https://doi.org/10.4236/ce.2019.102028
Best Regards, André
Andre Michaud,
Your linked paper omits its marvelous title, The Mechanics of Conceptual Thinking. The title is inviting, to say the least. I wrote a 2009 paper on a theory of intelligence, which is on arxiv and also RG, more about applying statistical mechanics to ensembles of problems solvers and ensembles of concepts. Though that seems to be a different take, I am motivated by the 2009 work to read your paper which I look forward to reading.
Best wishes.
Dear Robert,
I had a quick look at your project, and I see that you also relate language to intelligence. You should not be disappointed then as you read my paper, which is a synthesis of a line of research that definitely sets articulated languages as the foundation of intelligence and conceptual thinking.
If interested, these two older papers that are part of the same neurolinguistic project go into more specific details about the intelligence development aspects:
https://www.omicsonline.org/open-access/on-the-relation-between-the-comprehension-ability-and-the-neocortexverbal-areas-2155-6180-1000331.pdf
https://www.omicsonline.org/open-access/intelligence-and-early-mastery-of-the-reading-skill-2155-6180-1000327.pdf
Best Regards, André
RS: I am curious about how Euclid treated the issue of dimension.
I don't think he identified dimensions as a concept as we do but his books 1 to 6 deal only with geometry in 2 dimensions, books 7 to 10 deal with numbers and then books 11 to 13 extend book 6 to geometry in 3 dimensions.
George Dishman,
I am also curious about how Euclid treated dimension.
I have tried a beginning attempt to consider dimension including reference to Book X in Euclid's The Elements, in an article I posted this week on RG:
Preprint Euclidean commensurability, dimensional analysis and dimensi...
At best, I think it is a starting point.
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