Toward A Metaphysics of Nilpotent Regions Lu Chen University of Massachusetts, Amherst SMS 2019 1 / 27
Highlights Goal : Give a realistic interpretation of Smooth Infinitesimal Analysis (SIA) as a theory of space with infinitesimal features. Highlights : 1. Infinitesimals are widely used in heuristic reasoning in physics, and SIA can regiment such reasoning. 2. SIA is classically inconsistent and requires intuitionistic logic. It is commonly believed that there are no classical reconstructions of SIA. 3. I introduce a simple presheaf model for SIA in classical logic. 4. I give a realistic interpretation of the model, which involves a generalization of Einstein algebras in spacetime algebraicism. 2 / 27
Infinitesimals in Reasoning in Physics Suppose an object is moving at speed v in a circular orbit with radius r . What is the acceleration rate a ? v 1 and v 2 are two vector velocities at times infinitesimal apart. (Morin 2007, 98) ∆v = v 2 − v 1 ∆r = r 2 − r 1 | ∆v | / v = | ∆r | / r (triangle similarity) v · | ∆ v 1 ∆ t | = 1 r · | ∆ r ∆ t | a = v 2 / r 3 / 27
Infinitesimals in Geometry A circle is a regular polygon with infinitesimal sides. Radius= r OG = a Circumference= C Perimeter= P Area= 1 2 rC = π r 2 Area= 1 2 aP . 4 / 27
Smooth Infinitesimal Analysis Smooth infinitesimal analysis (SIA) is a theory of infinitesimals that purports to regiment those ideas. ◮ It can serve as an alternative foundation for calculus. ◮ In “Towards A Mathematics of Quantum Field Theory” (2012), Paugam aims to reformulate QFT in SIA. Microstraight (core claim) For any smooth curve, and any point on the curve, there is a straight infinitesimal segment of the curve around the point. cf. Robinson’s nonstandard analysis 5 / 27
Smooth infinitesimal analysis The smooth line R is a field that satisfies the following axiom (let ∆ = { x ∈ R | x 2 = 0 } ): Kock-Lawvere Axiom ( ∀ f : ∆ → R )( ∃ ! a , b ∈ R )( ∀ x ∈ ∆) f ( x ) = a + bx . 6 / 27
Nilpotent Infinitesimals ⇒ ¬∀ x ∈ R ( x 2 = 0 → x = 0). Microstraight = The set of nilpotent infinitesimals ∆ is not { 0 } . 7 / 27
No Non-Zero Nilpotent Infinitesimals Theorem There are no nilpotent infinitesimals that are not zero. Proof Suppose there is a non-zero nilpotent infinitesimal ǫ . ⇒ ǫ = ǫ · 1 = ǫ · ( ǫ · ǫ − 1 ) = ǫ 2 · ǫ − 1 = 0 · ǫ − 1 = 0 . R is a field = Contradiction. Therefore, there is no non-zero nilpotent infinitesimal. Note that the proof is both classically and intuitionistically valid. 8 / 27
Classical Inconsistency Two claims: 1 . ¬∀ x ∈ R ( x 2 = 0 → x = 0) . 2 . ¬∃ x ∈ R ( x 2 = 0 ∧ x � = 0) . Classical inconsistency . 2 implies ∀ x ∈ R ( x 2 = 0 → x = 0). Intuitionistic consistency. 2 intuitionistically implies ∀ x ∈ R ( x 2 = 0 → ¬ x � = 0). But, ¬ x � = 0 does not intuionistically imply x = 0. Note: 1 and 2 imply that the law of excluded middle is refuted in SIA. 9 / 27
Not Motivated by Constructivism Constructive Mathematics . Mathematical objects are constructions of the human mind and do not exist independently. 1 . ¬∀ x ∈ R ( x 2 = 0 → x = 0) = ⇒ We need to demonstrate it’s possible to construct an object that has a square of zero and is not equal to zero. 2 . ¬∃ x ∈ R ( x 2 = 0 ∧ x � = 0) = ⇒ It’s impossible to construct such an object. (Hellman 2006) 10 / 27
Indeterminacy of Identity? The idea: “=” means “determinately identical” “ � =” means “determinately not identical” (or “distinguishable”) = ⇒ ¬ ( ǫ = 0), ¬ ( ǫ � = 0). But, � = is not primitive, but a combination of ¬ and =. (Hellman 2006) 11 / 27
My Strategy Moerdijk and Reyes (1991) have constructed classical models for SIA using sheaf semantics . I will advance a shift of perspectives: From: A “Syntactic” View of SIA The sheaf models are only invoked to prove the consistency of SIA. (Bell 1998, Hellman 2006) To: A “Semantics” View of SIA The sheaf models are realistic representations of the world according to SIA. 12 / 27
The (Pre)Sheaf Model � W , D , C , v � W : quotient rings of C ∞ ( R ). D R : a presheaf (domain function) that assigns every w ∈ W the set of its ring elements. C : homomorphisms (counterpart maps) between members of W . For any w 1 , w 2 , if there is a homomorphism from w 1 to w 2 , then we say w 1 “sees” w 2 (abbr. w 1 Rw 2 ). 13 / 27
Examples of Quotient Rings of C ∞ ( R ) Quotient Rings of C ∞ ( R ) Equivalence classes of members of C ∞ ( R ) under certain equivalence relations that preserve the original ring structure Example 1. C ∞ ([0 , 1]) Two smooth functions on R are equivalent iff their values do not differ on [0 , 1] = ⇒ Isomorphic to all smooth functions on [0 , 1]. 14 / 27
The Ring of Linear Functions Example 2. The ring of linear functions L Two smooth functions on R are equivalent iff they have the same value at 0 and the same derivative at 0. = ⇒ Isomorphic to { f | f = a + bx } . Nilpotent Elements There are nilpotent ring elements that are not zero, e.g., f ( x ) = x . f � = 0 , but f 2 = 0 . 15 / 27
Interpreting SIA 1 . ¬∀ x ∈ R ( x 2 = 0 → x = 0) For any w 1 , and for any w 2 with w 1 Rw 2 , it is not the case that, for any w 3 with w 2 Rw 3 , for all d ∈ D R ( w 3 ), for any w 4 with a map h from w 3 to w 4 , if h ( d ) 2 = v w 4 (0), then h ( d ) = v w 4 (0). Put simply: every ring sees some ring that has non-zero nilpotent elements. A truth maker: L has non-zero nilpotent elements (e.g., f ( x ) = x ) and is accessible from every ring. 16 / 27
Interpreting SIA 2 . ¬∃ x ∈ R ( x 2 = 0 ∧ x � = 0) For any possible world w 1 , and for any possible world w 2 with w 1 Rw 2 , it is not the case that, there is a d ∈ D R ( w 2 ) such that d 2 = v w 2 (0) and for any possible world w 3 with a map h from w 2 to w 3 , h ( d ) � = v w 3 (0). Put simply: in any ring, every nilpotent element has zero as a counterpart in some ring. A truth maker: Every nilpotent element in every ring can be mapped to 0 in R . 17 / 27
A Realistic Interpretation The presheaf model consists of rings of smooth functions on standard space. = ⇒ SIA is actually a ring theory?? That wouldn’t help. We want some similarity between the interpretation of SIA and its intuitive meaning (a nonstandard theory of space). 18 / 27
Rings Represent Regions The rings actually represent regions of space, and homomorphisms between rings are maps and relations between their corresponding regions. 19 / 27
Einstein Algebras * Spacetime Algebraicism. Fields exist without an underlying spacetime. (Geroch 1972, Earman and Norton 1987) Manifold-Algebra Duality There is a one-to-one correspondence between manifolds and smooth algebras such that for any two manifolds M , N , every smooth map from M to N uniquely corresponds to a homomorphism from C ∞ ( N ) to C ∞ ( M ) and vice versa. (Rosenstock et al. 2015) 20 / 27
The “Geometric” Condition A ring is a smooth algebra only if it does not have non-zero nilpotent elements. However, without appealing to Manifold-Algebra Duality , this condition seems arbitrary and unmotivated. 21 / 27
Ring-Locus Duality Ring-Locus Duality There is a one-to-one correspondence between quotient rings of C ∞ ( R ) (more generally, of C ∞ ( R n )) and loci such that for any rings A , B , every homomorphism from A to B corresponds to a unique smooth map from B ’s corresponding locus to A ’s corresponding locus. Examples C ∞ ( R ) = the smooth line locus R L L = the nilpotent locus ∆ L . R = the point locus p . 22 / 27
Loci and SIA Loci exhibit some desirable features of SIA. SIA: ( ∀ f : ∆ → R )( ∃ ! a , b ∈ R )( ∀ x ∈ ∆) f ( x ) = a + bx . Loci: All smooth functions on ∆ L are linear. (“Smooth functions” on ∆ L are maps from ∆ L to R L , which are represented by homomorphisms from C ∞ ( R ) to L , which are isomorphic to L .) 23 / 27
Differentiation The derivative of a smooth function on R L on point p is the slope of the function restricting to ∆ p , the nilpotent region around p . ∆ L can be embedded as a nilpotent region of R L around any real number point through an injective map from ∆ L to R L . 24 / 27
Non-classical Mereology Supplementation. If x is a proper part of y , then y also has a proper part z that does not overlap x . Failure of Supplementation . The point locus p is a proper sublocus of ∆ L . But there is no other proper sublocus of ∆ L . 25 / 27
Conclusion I argue that the sheaf model for SIA proposed by Moerdijk and Reyes represents a geometric theory of loci with nilpotent regions that obeys classical logic. This theory is a generalization of Einstein algebra, can describe our actual space, and should be considered the real content of SIA. According to the theory, space has a non-classical mereology and, in particular, violates supplementation . 26 / 27
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