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Topic: [HM] geometric property of hyperbolic space
Replies: 1   Last Post: Aug 14, 2005 9:33 AM

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Irving Anellis

Posts: 15
Registered: 4/10/05
Re: [HM] geometric property of hyperbolic space
Posted: Aug 14, 2005 9:33 AM
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Quoting John Bell <>:

> Content preview: Can anybody clarify the following? In a 1891
> dictionary
> entry on Spinoza authored by C.S. Peirce, he writes that "The main
> principle [of the Ethics] is... an anticipation in a generalized
> form
> of the modern geometrical conception of the absolute, especially as
> this appears in the hyperbolic geometry, where the point and plane
> manifolds have a correspondence similar to that between Spinoza's
> worlds of extension and thought." [...]

In _Renati des Cartes principiorum philosophiae_ (1663), Spinoza
(1632-1677) sought to treat Descartes’ philosophy geometrically. And in
the _Tractatus de intellectus emendatione_ and the_Ethica, ordine
geometrico demonstrata_ (1675), he demanded a strong separation of the
Law of Contradiction from other logical connections. The Ethica is an
attempt to develop ethics axiomatically in the same respect that Euclid
in the _Elements_ developed geometry axiomatically.

The main thrust of Spinoza’s efforts in the Ethics (and his other
writings -- as I understand them) is to establish God as an infinite
absolute embracing through thought the the natural universe as an
extension of himself in space).

In particular, Spinoza wrote:

"By attribute I understand what the intellect perceives of a substance,
as constituting its essence"; "By God I understand a being absolutely
infinite, i.e., a substance consisting of an infinity of attributes, of
which each one expresses an eternal and infinite essence." The
definitions of Part One are, in effect, simply clear concepts that
ground the rest of his system. They are followed by a number of axioms
that, he assumes, will be regarded as obvious and unproblematic by the
philosophically informed ("Whatever is, is either in itself or in
another"; "From a given determinate cause the effect follows
necessarily"). From these, the first proposition necessarily
follows,and every subsequent proposition can be demonstrated using only
what precedes it. (References to the Ethics will be by part (I-V),
proposition (p), definition (d), scholium (s) and corollary (c).)

In propositions one through fifteen of Part One, Spinoza presents the
basic elements of his conception of God. God is the infinite,
necessarily existing (that is, uncaused), unique substance of the
universe. There is only one substance in the universe; it is God; and
everything else that is, is in God.

Proposition 1: A substance is prior in nature to its affections.

Proposition 2: Two substances having different attributes have nothing
in common with one another. (In other words, if two substances differ
in nature, then they have nothing in common).

Proposition 3: If things have nothing in common with one another, one of
them cannot be the cause of the other.

Proposition 4: Two or more distinct things are distinguished from one
another, either by a difference in the attributes [i.e., the natures or
essences] of the substances or by a difference in their affections
[i.e., their accidental properties].

Proposition 5: In nature, there cannot be two or more substances of the
same nature or attribute.

Proposition 6: One substance cannot be produced by another substance.

Proposition 7: It pertains to the nature of a substance to exist.

Proposition 8: Every substance is necessarily infinite.

Proposition 9: The more reality or being each thing has, the more
attributes belong to it.

Proposition 10: Each attribute of a substance must be conceived through

Proposition 11: God, or a substance consisting of infinite attributes,
each of which expresses eternal and infinite essence, necessarily
exists. (The proof of this proposition consists simply in the classic
"ontological proof for God's existence". Spinoza writes that "if you
deny this, conceive, if you can, that God does not exist. Therefore, by
axiom 7 [‘If a thing can be conceived as not existing, its essence does
not involve existence’], his essence does not involve existence. But
this, by proposition 7, is absurd. Therefore, God necessarily exists,

Proposition 12: No attribute of a substance can be truly conceived from
which it follows that the substance can be divided.

Proposition 13: A substance which is absolutely infinite is indivisible.

Proposition 14: Except God, no substance can be or be conceived.

As explained in the _Staford Encylopedia of Philosophy_:

"This proof that God, an infinite, necessary and uncaused, indivisible
being, is the only substance of the universe proceeds in three simple
steps. First, establish that no two substances can share an attribute
or essence . Then, prove that there is a substance with infinite
attributes (i.e., God). It follows, in conclusion, that the existence
of that infinite substance precludes the existence of any other
substance. For if there were to be a second substance, it would have to
have some attribute or essence. But since God has all possible
attributes, then the attribute to be possessed by this second substance
would be one of the attributes already possessed by God. But it has
already been established that no two substances can have the same
attribute. Therefore, there can be, besides God, no such second

"The two attributes of God of which we have cognizance are extension and
thought. This, in itself, involves what would have been an astounding
thesis in the eyes of his contemporaries, one that was usually
misunderstood and always vilified. When Spinoza claims in Proposition
Two that "Extension is an attribute of God, or God is an extended
thing", he was almost universally -- but erroneously -- interpreted as
saying that God is literally corporeal. For just this reason,
"Spinozism" became, for his critics, synonymous with atheistic

"According to one interpretation, God is indeed material, even matter
itself, but this does not imply that God has a body. Another
interpretation, however, one which will be adopted here, is that what
is in God is not matter per se, but extension as an essence. And
extension and thought are two distinct essences that have absolutely
nothing in common. The modes or expressions of extension are physical
bodies; the modes of thought are ideas. Because extension and thought
have nothing in common, the two realms of matter and mind are causally
closed systems. Everything that is extended follows from the attribute
of extension alone. Every bodily event is part of an infinite causal
series of bodily events and is determined only by the nature
ofextension and its laws, in conjunction with its relations to other
extended bodies. Similarly, every idea follows only from the attribute
of thought. Any idea is an integral part of an infinite series of ideas
and is determined by the nature of thought and its laws, along with its
relations to other ideas. There is, in other words, no causal
interaction between bodies and ideas, between the physical and the
mental. There is, however, a thoroughgoing correlation and parallelism
between the two series. For every mode in extension that is a
relatively stable collection of matter, there is a corresponding mode
in thought. In fact, he insists, "a mode of extension and the idea of
that mode are one and the same thing, but expressed in two ways".
Because of the fundamental and underlying unity of Nature, or of
Substance, Thought and Extension are just two different ways of
"comprehending" one and the same Nature. Every material thing thus has
its own particular idea -- an eternal adequate idea -- that expresses
or represents it. Since that idea is just a mode of one of God's
attributes -- Thought -- it is in God, and the infinite series of ideas
constitutes God's mind or infinite intellect. As he explains,

"A circle existing in nature and the idea of the existing circle, which
is also in God, are one and the same thing, which is explained through
different attributes. Therefore, whether we conceive nature under the
attribute of Extension, or under the attribute of Thought, or under any
other attribute, we shall find one and the same order, or one and the
same connection of causes, i.e., that the same things follow one

"It follows from this, he argues, that the causal relations between
bodies is mirrored in the logical relations between God's ideas. Or, as
Spinoza notes in Proposition Seven, "the order and connection of ideas
is the same as the order and connection of things".

"One kind of extended body, however, is significantly more complex than
any others in its composition and in its dispositions to act and be
acted upon. That complexity is reflected in its corresponding idea. The
body in question is the human body; and its corresponding idea is the
human mind or soul. The mind, then, like any other idea, is simply one
particular mode of God's attribute, Thought. Whatever happens in the
body is reflected or expressed in the mind. In this way, the mind
perceives, more or less obscurely, what is taking place in its body.
And through its body's interactions with other bodies, the mind is
aware of what is happening in the physical world around it. But the
human mind no more interacts with its body than any mode of Thought
interacts with a mode of Extension."

Among the ways in which things depend upon God, are included "the most
general laws of the universe, together governing all things in all
ways. From the attribute of extension there follow the principles
governing all extended objects (the truths of geometry) and laws
governing the motion and rest of bodies (the laws of physics); from the
attribute of thought, there follow laws of thought (understood by
commentators to be either the laws of logic or the laws of

For background and exposition, see esp. Edwin M. Curley, _Behind the
Geometrical Method: A Reading of Spinoza's Ethics_ (Princeton:
Princeton University Press, 1988), and Aaron V. Garrett, _Meaning in
Spinoza's Method_ (Cambridge/New York: Cambridge University Press,

Peirce produced a number of unpublished manuscripts on non-Euclidean
geometry; the following is the list from the standard catalog of Peirce
writings, published and unpublished (they are manuscripts Robin catalog
## 114-124, as follows)

114. On Hyperbolic Geometry (Hyp. Geom)
A. MS., n.p., [c.1901?], pp. 1-6, 16-20, with rejected pages.
Formulae required for the projection of the hyperbolic plane upon the
Euclidean. Definitions of "individual," "independence of individuals,"
and "collection." Fundamental theorem of multitude. (Cantor's
demonstration of this theorem is thought to be fallacious.)

115. Newton's Enumeration of Cubic Curves
A. MS., n.p., n.d., 7 pp.
Hyperbolic geometry.

116. Brocardian Geometry
A. MS., n.p., n.d., 1 p.

117. The Non-Euclidean Geometry made Easy
A. MS., G-undated-7, pp. 1-8.
Published, in part, as 8.97-99. Unpublished (pp. 3-8). Denial of either
the first or second of the two "natural propositions," noted in that
part of manuscript which was published, leads to a non-Euclidean
geometry. Both of the corresponding kinds of non-Euclidean geometry are
intelligible, and a consideration of plane geometry will suffice to show

118. Reflections on Non-Euclidean Geometry
A. MS., n.p., n.d., pp. 1-5.

119. Non-Euclidean Geometry
A. MS., n.p., [c.1883 or later], 1 p. and 1 p. ("Notes on Non-Euclidean
Geometry") .
The purpose of this memoir is to find some way of treating geometry
metrically by introducing the absolute synthetically. The attempt is
restricted to plane non-Euclidean geometry: "Solid non-Euclidean
geometry is a trifle too hard for me."

120. The Elements of Non-Euclidean Geometry. Preface
A. MS., n.p., n.d., 3 pp., plus 3 pp. which may be part of the same

121. [On Non-Euclidean Geometry]
A. MS., G-undated-6, pp. 2-11; plus 4 pp. of an earlier draft.
Probably manuscript of an address to the New York Mathematical Society,
November 24, 1894. Published, in part, as 8.93 n2. Was Euclid a
non-Euclidean geometer? Probably! Properties of space. Evidence for
thinking there is an absolute which is a real quadric surface. Newton's
argument that space is an entity and its bearing on non-Euclidean
Geometry. On back of p. 11: "Professor Fiske" [i.e., Thomas S. Fiske].

122. Non-Euclidean Geometry. Sketch of a Synthetic Treatment
A. MS., n.p., n.d., 32 pp. (several attempts with different titles).

123. Lobachevski's Geometry
A. MS., n.p., n.d., 3 pp.

124. Formulae
A. MS., notebook, n.p., n.d.
Notes on non-Euclidean geometry, existential graphs, and Laurent's
probabilities. Solution of quadratic equation. The "formulae" of the
title refers to trigonometrical formulae and formulae of analytic

Indubitably the best resource to obtain an informed answer your question
would be on the Peirce discussion group, which you can join by sending a
subscribe PEIRCE-L message to:

Irving H. Anellis

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