Toward A Metaphysics of Nilpotent Regions Lu Chen University of - - PowerPoint PPT Presentation

Toward A Metaphysics of Nilpotent Regions

Lu Chen

University of Massachusetts, Amherst

SMS 2019

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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

• f 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.

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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? v1 and v2 are two vector velocities at times infinitesimal

• apart. (Morin 2007, 98)

∆v = v2−v1 ∆r = r2−r1 |∆v|/v = |∆r|/r (triangle similarity)

1 v · | ∆v ∆t | = 1 r · | ∆r ∆t |

a = v2/r

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Infinitesimals in Geometry

A circle is a regular polygon with infinitesimal sides. OG = a Perimeter= P Area= 1

2aP.

Radius= r Circumference= C Area= 1

2rC = πr2

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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

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Smooth infinitesimal analysis

The smooth line R is a field that satisfies the following axiom (let ∆ = {x ∈ R | x2 = 0}):

Kock-Lawvere Axiom

(∀f : ∆ → R)(∃!a, b ∈ R)(∀x ∈ ∆)f (x) = a + bx.

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Nilpotent Infinitesimals

Microstraight = ⇒ ¬∀x ∈ R(x2 = 0 → x = 0). The set of nilpotent infinitesimals ∆ is not {0}.

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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. R is a field = ⇒ ǫ = ǫ · 1 = ǫ · (ǫ · ǫ−1) = ǫ2 · ǫ−1 = 0 · ǫ−1 = 0. 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(x2 = 0 → x = 0). 2.¬∃x ∈ R(x2 = 0 ∧ x = 0). Classical inconsistency. 2 implies ∀x ∈ R(x2 = 0 → x = 0). Intuitionistic consistency. 2 intuitionistically implies ∀x ∈ R(x2 = 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

• f the human mind and do not exist independently.

1.¬∀x ∈ R(x2 = 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(x2 = 0 ∧ x = 0) = ⇒ It’s impossible to construct such an object. (Hellman 2006)

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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)

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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.

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The (Pre)Sheaf Model

W , D, C, v W : quotient rings of C ∞(R). DR: 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 w1, w2, if there is a homomorphism from w1 to w2, then we say w1 “sees” w2 (abbr. w1Rw2).

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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].

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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.

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Interpreting SIA

1.¬∀x ∈ R(x2 = 0 → x = 0)

For any w1, and for any w2 with w1Rw2, it is not the case that, for any w3 with w2Rw3, for all d ∈ DR(w3), for any w4 with a map h from w3 to w4, if h(d)2 = vw4(0), then h(d) = vw4(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.

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Interpreting SIA

2.¬∃x ∈ R(x2 = 0 ∧ x = 0)

For any possible world w1, and for any possible world w2 with w1Rw2, it is not the case that, there is a d ∈ DR(w2) such that d2 = vw2(0) and for any possible world w3 with a map h from w2 to w3, h(d) = vw3(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.

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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).

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Rings Represent Regions

The rings actually represent regions of space, and homomorphisms between rings are maps and relations between their corresponding regions.

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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)

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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.

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Ring-Locus Duality

Ring-Locus Duality

There is a one-to-one correspondence between quotient rings of C ∞(R) (more generally, of C ∞(Rn)) 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 RL L = the nilpotent locus ∆L. R = the point locus p.

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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 RL, which are represented by homomorphisms from C ∞(R) to L, which are isomorphic to L.)

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Differentiation

The derivative of a smooth function on RL on point p is the slope

• f the function restricting to ∆p, the nilpotent region around p.

∆L can be embedded as a nilpotent region of RL around any real number point through an injective map from ∆L to RL.

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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.

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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.

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Presheaf Semantics

• Negation. vw(¬p) = 1 iff for all w′ such that wRw′, vw′(p) = 0.
• Conjunction. vw(p ∧ q) = 1 iff vw(p) = 1 and vw(q) = 1.
• Disjunction. vw(p ∨ q) = 1 iff vw(p) = 1 or vw(q) = 1.
• Conditional. vw(p → q) = 1 iff for all w′ such that wRw′, if

vw′(p) = 1, then vw′(q) = 1. Universal quantifier. vw(∀xφ(x)) = 1 iff for all w′ such that wRw′, and for all d ∈ X(w′), d satisfies vw′(φ). Existential quantifier. vw(∃xφ(x)) = 1 iff there is a d ∈ X(w), d satisfies vw(φ).

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