I am planning to read Spinoza's Ethics, Geometrically Demonstrated, and before I read a work I peruse throughout the work, keeping note of headings, as well as read the table of contents, to get a general idea of the structure of the work. The system of axioms and propositions was very similar to Euclid's work on geometry, as well as Newton's Principia Mathematica. The name ("geometrically demonstrated"), implies that this structure was intentional and is derived from Euclid's work.

What struck me about this structure was that I had not found it while reading Descarte, Hume or other philosopher's works. Most works, however structured, seem to be written more in an essay like format rather than being composed of axioms and propositions with the occasional scholium.

My question, thus, is why does Spinoza "geometrically demonstrate" his philosophical worl ('ethics), i.e. why does he structure his work like Euclids? Is my conclusion that he structured it intentionally, which appears to be confirmed by wikipedia (a recourse which is not very credible according to my teachers), correct? Does he do it to serve a philosophical purpose, because it was the style of the time, or for a completely different reason?

Note: I haven't yet started read the Ethics, Geometrically Demonstrated, (having just perused the work), so if this question is elucidated by a reading, then please let me know (esp. if there is a relevant section), though in additon please explain or summarize the reasons why in additon.

  • 2
    I'm not a Spinoza scholar, but I guess that he wanted for his philosophical work to achieve the same logical rigor as that of Euclid and the mathematicians.
    – Adrian
    May 25, 2015 at 21:00
  • Interesting idea @Nicol. However, Descarte himself in his Meditations, in the first part addressed to the scared faculty of theology at Paris, himself states that he provides (philosophical and theological) arguments with clarity meeting or exceeding those of geometry. However, he writes in a more standard essay-like, prose form instead of fashioning his work on Euclid's. Hence, considering that Spinoza came after Descarte, in addition to a desire to achieve a logical rigor matching Euclid's geometry, perhaps Spinoza had an additional goal which Descarte didn't have.
    – Cicero
    May 25, 2015 at 21:07

2 Answers 2


Look at how the Ethics bids farewell to its reader:

If the way which I have pointed out as leading to this result seems exceedingly hard, it may nevertheless be discovered. Needs must it be hard, since it is so seldom found. How would it be possible, if salvation were ready to our hand, and could without great labour be found, that it should be by almost all men neglected? But all things excellent are as difficult as they are rare.

The choice of style reflects the need for Spinoza to imply his reader as deeply as he could into the seriousness of the matter at hand. Not just a search for precision, but a plea (and a demand) of austerity. The Ethics is not a commentary(!) and what is more distant from commentary than a mathematical demonstration?

The work is a serious and direct challenge to the powers of its time. As little as possible could be left to chance, and this had to be made explicit to a reader that was being invited to a risky journey, possibly without return.

  • So because of the revolutionary nature of Spinoza's work (unlike the more conservative nature of Descartes', a conservative christian), he had to make sure people would take his notions seriously. Hence, he created a mathematical demonstration to put his ideas in exact form, to firstly achieve logical rigor but more importantly to make his work be taken seriously. Thanks, @AndreSouzaLemos
    – Cicero
    May 25, 2015 at 23:01
  • You could say that. May 25, 2015 at 23:02

This response to why Spinoza chose a Geometric Structure will be new and appear awkward at first read. But given time it may result in a realization that Spinoza 'anticipated' the effects of 'Human Geometry' long before that thought ever entered my mind.

Almost 40 years ago, upon stumbling across a statement by Sir Thomas Heath [1861-1940], in The History of Mathematics, he who was considered by many to be the truly greatest historian in the field of mathematics, noted that; “…mathematics from its origination up until the time of the early Greek mathematicians did not involve any numbers”. Shortly after reading this a process of imagining just how a mathematics without numbers would originate and how it would operate began and soon, without any prompting, began to grow into a realization. An awakening began to germinate and eventually to blossom into exactly why Spinoza selected the ‘more geometrica’ and why his reason must have sprung from his intuition of a faculty innate within us, a faculty which prompts its own moniker; ‘human geometry’. What Baruch is replicating in his use of the Geometric Method is the same process involved in any geometry problem where the geometric figure, whether a square, a circle, etc. serves as one half of a complete demonstration. The other half is comprised of the hypotheses, postulates, corollaries etc. which complete the demonstration. In Spinoza's case the first half of his demonstration is comprised of his axioms, propositions, corollaries, lemmas and scholia. The other half is the human mind which stands in for the figure in plane geometry and effectively 'completes' the demonstration.

And here is why the analogy between a Euclidean demonstration and a Spinozan one holds true and why Baruch selected Geometry as his medium of expression. As a human Geometer each of us contains inside of us a gyroscopic sensor which automatically guides us in our circumnavigation of our environment. This amounts to the recognition that a sensation, barely perceptible but real, not unlike invisible tentacles reaches out from inside of each of us acting like a GPS/Radar; groping and wrapping itself around the objects in our world which surround us; tentacles which lash us firmly to our extended environment and provide us with a magnetic compass like attachment and make possible our true north and guiding star for navigating our way through life.

For more on this 'human geometry and the 'more geometrica; visit charlessaunders5.academia.edu. Download 'To Discern Divinity' for free and see pp 20-45. Regards, CS

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