Reuniting the Antipodes: Bringing together Nonstandard and - - PowerPoint PPT Presentation

▶
reuniting the antipodes
SMART_READER_LITE
LIVE PREVIEW

Reuniting the Antipodes: Bringing together Nonstandard and - - PowerPoint PPT Presentation

Aug 30, 2022 254 likes •2.09k views

Reuniting the Antipodes: Bringing together Nonstandard and Constructive Analysis Sam Sanders 1 Madison, WI, April 2, 2012 1 This research is generously supported by the John Templeton Foundation. Introduction NSA and Constructive Analysis

slide-1
SLIDE 1

Reuniting the Antipodes:

Bringing together Nonstandard and Constructive Analysis

Sam Sanders1 Madison, WI, April 2, 2012

1This research is generously supported by the John Templeton Foundation.

slide-2
SLIDE 2

Introduction NSA and Constructive Analysis Philosophical implications

Motivation

From the scope of CCA 2012: (http://cca-net.de/cca2012/)

slide-3
SLIDE 3

Introduction NSA and Constructive Analysis Philosophical implications

Motivation

From the scope of CCA 2012: (http://cca-net.de/cca2012/) The conference is concerned with the theory of computability and complexity over real-valued data.

slide-4
SLIDE 4

Introduction NSA and Constructive Analysis Philosophical implications

Motivation

From the scope of CCA 2012: (http://cca-net.de/cca2012/) The conference is concerned with the theory of computability and complexity over real-valued data. [BECAUSE]

slide-5
SLIDE 5

Introduction NSA and Constructive Analysis Philosophical implications

Motivation

From the scope of CCA 2012: (http://cca-net.de/cca2012/) The conference is concerned with the theory of computability and complexity over real-valued data. [BECAUSE] Most mathematical models in physics and engineering [...] are based on the real number concept.

slide-6
SLIDE 6

Introduction NSA and Constructive Analysis Philosophical implications

Motivation

From the scope of CCA 2012: (http://cca-net.de/cca2012/) The conference is concerned with the theory of computability and complexity over real-valued data. [BECAUSE] Most mathematical models in physics and engineering [...] are based on the real number concept. The following is more true: Most mathematical models in physics and engineering [...] are based on the real number concept, and an intuitive calculus with infinitesimals, i.e. informal Nonstandard Analysis.

slide-7
SLIDE 7

Introduction NSA and Constructive Analysis Philosophical implications

Motivation

From the scope of CCA 2012: (http://cca-net.de/cca2012/) The conference is concerned with the theory of computability and complexity over real-valued data. [BECAUSE] Most mathematical models in physics and engineering [...] are based on the real number concept. The following is more true: Most mathematical models in physics and engineering [...] are based on the real number concept, and an intuitive calculus with infinitesimals, i.e. informal Nonstandard Analysis. We present a notion of constructive computability directly based

  • n Nonstandard Analysis.
slide-8
SLIDE 8

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Analysis

Errett Bishop’s Constructive Analysis is a redevelopment of Mathematics, consistent with CLASS, RUSS and INT.

slide-9
SLIDE 9

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Analysis

Errett Bishop’s Constructive Analysis is a redevelopment of Mathematics, consistent with CLASS, RUSS and INT.

Definition (Logical connectives in BISH: BHK)

1

P ∨ Q: we have an algorithm that outputs either P or Q, together with a proof of the chosen disjunct.

slide-10
SLIDE 10

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Analysis

Errett Bishop’s Constructive Analysis is a redevelopment of Mathematics, consistent with CLASS, RUSS and INT.

Definition (Logical connectives in BISH: BHK)

1

P ∨ Q: we have an algorithm that outputs either P or Q, together with a proof of the chosen disjunct.

2

P ∧ Q: we have both a proof of P and a proof of Q.

slide-11
SLIDE 11

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Analysis

Errett Bishop’s Constructive Analysis is a redevelopment of Mathematics, consistent with CLASS, RUSS and INT.

Definition (Logical connectives in BISH: BHK)

1

P ∨ Q: we have an algorithm that outputs either P or Q, together with a proof of the chosen disjunct.

2

P ∧ Q: we have both a proof of P and a proof of Q.

3

P → Q: by means of an algorithm we can convert any proof of P into a proof of Q.

slide-12
SLIDE 12

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Analysis

Errett Bishop’s Constructive Analysis is a redevelopment of Mathematics, consistent with CLASS, RUSS and INT.

Definition (Logical connectives in BISH: BHK)

1

P ∨ Q: we have an algorithm that outputs either P or Q, together with a proof of the chosen disjunct.

2

P ∧ Q: we have both a proof of P and a proof of Q.

3

P → Q: by means of an algorithm we can convert any proof of P into a proof of Q.

4

¬P ≡ P → (0 = 1).

slide-13
SLIDE 13

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Analysis

Errett Bishop’s Constructive Analysis is a redevelopment of Mathematics, consistent with CLASS, RUSS and INT.

Definition (Logical connectives in BISH: BHK)

1

P ∨ Q: we have an algorithm that outputs either P or Q, together with a proof of the chosen disjunct.

2

P ∧ Q: we have both a proof of P and a proof of Q.

3

P → Q: by means of an algorithm we can convert any proof of P into a proof of Q.

4

¬P ≡ P → (0 = 1).

5

(∃x)P(x): an algorithm computes an object x0 such that P(x0)

slide-14
SLIDE 14

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Analysis

Errett Bishop’s Constructive Analysis is a redevelopment of Mathematics, consistent with CLASS, RUSS and INT.

Definition (Logical connectives in BISH: BHK)

1

P ∨ Q: we have an algorithm that outputs either P or Q, together with a proof of the chosen disjunct.

2

P ∧ Q: we have both a proof of P and a proof of Q.

3

P → Q: by means of an algorithm we can convert any proof of P into a proof of Q.

4

¬P ≡ P → (0 = 1).

5

(∃x)P(x): an algorithm computes an object x0 such that P(x0)

6

(∀x ∈ A)P(x): for all x, x ∈ A → P(x).

slide-15
SLIDE 15

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Analysis

Errett Bishop’s Constructive Analysis is a redevelopment of Mathematics, consistent with CLASS, RUSS and INT.

Definition (Logical connectives in BISH: BHK)

1

P ∨ Q: we have an algorithm that outputs either P or Q, together with a proof of the chosen disjunct.

2

P ∧ Q: we have both a proof of P and a proof of Q.

3

P → Q: by means of an algorithm we can convert any proof of P into a proof of Q.

4

¬P ≡ P → (0 = 1).

5

(∃x)P(x): an algorithm computes an object x0 such that P(x0)

6

(∀x ∈ A)P(x): for all x, x ∈ A → P(x).

slide-16
SLIDE 16

Introduction NSA and Constructive Analysis Philosophical implications

From BISH to NSA

In BISH, proof and algorithm are central.

slide-17
SLIDE 17

Introduction NSA and Constructive Analysis Philosophical implications

From BISH to NSA

In BISH, proof and algorithm are central. We will define a system of Nonstandard Analyis NSA where transfer and Ω-invariant procedure play the same role.

slide-18
SLIDE 18

Introduction NSA and Constructive Analysis Philosophical implications

From BISH to NSA

In BISH, proof and algorithm are central. We will define a system of Nonstandard Analyis NSA where transfer and Ω-invariant procedure play the same role. NSA is similar to ∗RCA0, a nonstandard extension of RCA0.

slide-19
SLIDE 19

Introduction NSA and Constructive Analysis Philosophical implications

From BISH to NSA

In BISH, proof and algorithm are central. We will define a system of Nonstandard Analyis NSA where transfer and Ω-invariant procedure play the same role. NSA is similar to ∗RCA0, a nonstandard extension of RCA0. Three important features:

slide-20
SLIDE 20

Introduction NSA and Constructive Analysis Philosophical implications

From BISH to NSA

In BISH, proof and algorithm are central. We will define a system of Nonstandard Analyis NSA where transfer and Ω-invariant procedure play the same role. NSA is similar to ∗RCA0, a nonstandard extension of RCA0. Three important features:

1 No transfer principle / elementary extension (except for ∆0).

slide-21
SLIDE 21

Introduction NSA and Constructive Analysis Philosophical implications

From BISH to NSA

In BISH, proof and algorithm are central. We will define a system of Nonstandard Analyis NSA where transfer and Ω-invariant procedure play the same role. NSA is similar to ∗RCA0, a nonstandard extension of RCA0. Three important features:

1 No transfer principle / elementary extension (except for ∆0). 2 No ∆1-CA, but Ω-CA.

slide-22
SLIDE 22

Introduction NSA and Constructive Analysis Philosophical implications

From BISH to NSA

In BISH, proof and algorithm are central. We will define a system of Nonstandard Analyis NSA where transfer and Ω-invariant procedure play the same role. NSA is similar to ∗RCA0, a nonstandard extension of RCA0. Three important features:

1 No transfer principle / elementary extension (except for ∆0). 2 No ∆1-CA, but Ω-CA. 3 Levels of infinity (Stratified NSA).

slide-23
SLIDE 23

Introduction NSA and Constructive Analysis Philosophical implications

Feature 3: Stratified Nonstandard Analysis

slide-24
SLIDE 24

Introduction NSA and Constructive Analysis Philosophical implications

Feature 3: Stratified Nonstandard Analysis

The usual picture of ∗N:

∗N, the hypernatural numbers

  • . . .

ω2 . . . ωk . . .

✲

ω1 0 1 . . .

  • N, the finite numbers
  • Ω = ∗N \ N, the infinite numbers
slide-25
SLIDE 25

Introduction NSA and Constructive Analysis Philosophical implications

Feature 3: Stratified Nonstandard Analysis

The usual picture of ∗N:

∗N, the hypernatural numbers

  • . . .

ω2 . . . ωk . . .

✲

ω1 0 1 . . .

  • N, the finite numbers
  • Ω = ∗N \ N, the infinite numbers

In NSA, ∗N has extra structure: countably many levels of infinity N ⊂ N1 ⊂ . . . Nk ⊂ Nk+1 ⊂ · · · ⊂ ∗N

slide-26
SLIDE 26

Introduction NSA and Constructive Analysis Philosophical implications

Feature 3: Stratified Nonstandard Analysis

The usual picture of ∗N:

∗N, the hypernatural numbers

  • . . .

ω2 . . . ωk . . .

✲

ω1 0 1 . . .

  • N, the finite numbers
  • Ω = ∗N \ N, the infinite numbers

In NSA, ∗N has extra structure: countably many levels of infinity N ⊂ N1 ⊂ . . . Nk ⊂ Nk+1 ⊂ · · · ⊂ ∗N

N1, the 1-finite numbers

  • Ω1=∗N\N1, the 1-infinite numbers
slide-27
SLIDE 27

Introduction NSA and Constructive Analysis Philosophical implications

Feature 3: Stratified Nonstandard Analysis

The usual picture of ∗N:

∗N, the hypernatural numbers

  • . . .

ω2 . . . ωk . . .

✲

ω1 0 1 . . .

  • N, the finite numbers
  • Ω = ∗N \ N, the infinite numbers

In NSA, ∗N has extra structure: countably many levels of infinity N ⊂ N1 ⊂ . . . Nk ⊂ Nk+1 ⊂ · · · ⊂ ∗N

N2, the 2-finite numbers

  • Ω2=∗N\N2, the 2-infinite numbers
slide-28
SLIDE 28

Introduction NSA and Constructive Analysis Philosophical implications

Feature 3: Stratified Nonstandard Analysis

The usual picture of ∗N:

∗N, the hypernatural numbers

  • . . .

ω2 . . . ωk . . .

✲

ω1 0 1 . . .

  • N, the finite numbers
  • Ω = ∗N \ N, the infinite numbers

In NSA, ∗N has extra structure: countably many levels of infinity N ⊂ N1 ⊂ . . . Nk ⊂ Nk+1 ⊂ · · · ⊂ ∗N

Nk, the k-finite numbers

  • Ωk=∗N\Nk, the k-infinite numbers
slide-29
SLIDE 29

Introduction NSA and Constructive Analysis Philosophical implications

Feature 2: Ω-invariance

slide-30
SLIDE 30

Introduction NSA and Constructive Analysis Philosophical implications

Feature 2: Ω-invariance

Ω-invariance ≈ algorithm ≈ finite procedure ≈ explicit computation.

slide-31
SLIDE 31

Introduction NSA and Constructive Analysis Philosophical implications

Feature 2: Ω-invariance

Ω-invariance ≈ algorithm ≈ finite procedure ≈ explicit computation.

Definition (Ω-invariance)

A set A ⊂ N is Ω-invariant

slide-32
SLIDE 32

Introduction NSA and Constructive Analysis Philosophical implications

Feature 2: Ω-invariance

Ω-invariance ≈ algorithm ≈ finite procedure ≈ explicit computation.

Definition (Ω-invariance)

A set A ⊂ N is Ω-invariant if there is a quantifier-free formula ψ such that for all ω ∈ Ω,

slide-33
SLIDE 33

Introduction NSA and Constructive Analysis Philosophical implications

Feature 2: Ω-invariance

Ω-invariance ≈ algorithm ≈ finite procedure ≈ explicit computation.

Definition (Ω-invariance)

A set A ⊂ N is Ω-invariant if there is a quantifier-free formula ψ such that for all ω ∈ Ω, A = {k ∈ N : ψ(k, ω)}.

slide-34
SLIDE 34

Introduction NSA and Constructive Analysis Philosophical implications

Feature 2: Ω-invariance

Ω-invariance ≈ algorithm ≈ finite procedure ≈ explicit computation.

Definition (Ω-invariance)

A set A ⊂ N is Ω-invariant if there is a quantifier-free formula ψ such that for all ω ∈ Ω, A = {k ∈ N : ψ(k, ω)}.

Note that A depends on ω ∈ Ω, but not on the choice of ω ∈ Ω.

slide-35
SLIDE 35

Introduction NSA and Constructive Analysis Philosophical implications

Feature 2: Ω-invariance

Theorem (Finiteness)

For every Ω-invariant A ⊂ N and k ∈ N, there is M ∈ N, such that

slide-36
SLIDE 36

Introduction NSA and Constructive Analysis Philosophical implications

Feature 2: Ω-invariance

Theorem (Finiteness)

For every Ω-invariant A ⊂ N and k ∈ N, there is M ∈ N, such that k ∈ A ⇐ ⇒ ψ(k, ω) ⇐ ⇒ ψ(k, M).

slide-37
SLIDE 37

Introduction NSA and Constructive Analysis Philosophical implications

Feature 2: Ω-invariance

Theorem (Finiteness)

For every Ω-invariant A ⊂ N and k ∈ N, there is M ∈ N, such that k ∈ A ⇐ ⇒ ψ(k, ω) ⇐ ⇒ ψ(k, M).

Thus, to verify whether k ∈ A, we only need to perform finitely many

  • perations (i.e. determine if ψ(k, M)).
slide-38
SLIDE 38

Introduction NSA and Constructive Analysis Philosophical implications

Feature 2: Ω-invariance

Theorem (Finiteness)

For every Ω-invariant A ⊂ N and k ∈ N, there is M ∈ N, such that k ∈ A ⇐ ⇒ ψ(k, ω) ⇐ ⇒ ψ(k, M).

Thus, to verify whether k ∈ A, we only need to perform finitely many

  • perations (i.e. determine if ψ(k, M)).

NSA has Ω-CA instead of ∆1-CA.

Principle (Ω-CA)

All Ω-invariant sets exist.

slide-39
SLIDE 39

Introduction NSA and Constructive Analysis Philosophical implications

Lost in translation

slide-40
SLIDE 40

Introduction NSA and Constructive Analysis Philosophical implications

Lost in translation

BISH (based on IL)

slide-41
SLIDE 41

Introduction NSA and Constructive Analysis Philosophical implications

Lost in translation

BISH (based on IL) NSA (based on CL)

slide-42
SLIDE 42

Introduction NSA and Constructive Analysis Philosophical implications

Lost in translation

BISH (based on IL) NSA (based on CL)

Central: algorithm and proof

slide-43
SLIDE 43

Introduction NSA and Constructive Analysis Philosophical implications

Lost in translation

BISH (based on IL) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B

slide-44
SLIDE 44

Introduction NSA and Constructive Analysis Philosophical implications

Lost in translation

BISH (based on IL) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and transfer (T)

slide-45
SLIDE 45

Introduction NSA and Constructive Analysis Philosophical implications

Lost in translation

BISH (based on IL) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and transfer (T)

A V B: [A ∨ B] ∧ [A → A ∈ T] ∧ [B → B ∈ T]

slide-46
SLIDE 46

Introduction NSA and Constructive Analysis Philosophical implications

Lost in translation

BISH (based on IL) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and transfer (T)

A V B: [A ∨ B] ∧ [A → A ∈ T] ∧ [B → B ∈ T] A → B: an algo converts a proof of A to a proof of B

slide-47
SLIDE 47

Introduction NSA and Constructive Analysis Philosophical implications

Lost in translation

BISH (based on IL) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and transfer (T)

A V B: [A ∨ B] ∧ [A → A ∈ T] ∧ [B → B ∈ T] A → B: an algo converts a proof of A to a proof of B A ⇛ B: A ∧ [A ∈ T] → B ∧ [B ∈ T]

slide-48
SLIDE 48

Introduction NSA and Constructive Analysis Philosophical implications

Lost in translation

BISH (based on IL) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and transfer (T)

A V B: [A ∨ B] ∧ [A → A ∈ T] ∧ [B → B ∈ T] A → B: an algo converts a proof of A to a proof of B A ⇛ B: A ∧ [A ∈ T] → B ∧ [B ∈ T]

‘A ∈ T’ means ‘A satisfies Transfer’.

slide-49
SLIDE 49

Introduction NSA and Constructive Analysis Philosophical implications

Lost in translation

BISH (based on IL) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and transfer (T)

A V B: [A ∨ B] ∧ [A → A ∈ T] ∧ [B → B ∈ T] A → B: an algo converts a proof of A to a proof of B A ⇛ B: A ∧ [A ∈ T] → B ∧ [B ∈ T]

‘A ∈ T’ means ‘A satisfies Transfer’. E.g. (∀n ∈ N)ϕ(n) ∈ T is [(∀n ∈ N)ϕ(n) → (∀n ∈ ∗N)ϕ(n)]

slide-50
SLIDE 50

Introduction NSA and Constructive Analysis Philosophical implications

Lost in translation

BISH (based on IL) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and transfer (T)

A V B: [A ∨ B] ∧ [A → A ∈ T] ∧ [B → B ∈ T] A → B: an algo converts a proof of A to a proof of B A ⇛ B: A ∧ [A ∈ T] → B ∧ [B ∈ T]

‘A ∈ T’ means ‘A satisfies Transfer’. E.g. (∀n ∈ N)ϕ(n) ∈ T is [(∀n ∈ N)ϕ(n) → (∀n ∈ ∗N)ϕ(n)] E.g. (∃n ∈ ∗N)ϕ(n) ∈ T is [(∃n ∈ ∗N)ϕ(n) → (∃n ∈ N1)ϕ(n)]

slide-51
SLIDE 51

Introduction NSA and Constructive Analysis Philosophical implications

Lost in translation

BISH (based on IL) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and transfer (T)

A V B: [A ∨ B] ∧ [A → A ∈ T] ∧ [B → B ∈ T] A → B: an algo converts a proof of A to a proof of B A ⇛ B: A ∧ [A ∈ T] → B ∧ [B ∈ T]

¬A: A → (0 = 1)

slide-52
SLIDE 52

Introduction NSA and Constructive Analysis Philosophical implications

Lost in translation

BISH (based on IL) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and transfer (T)

A V B: [A ∨ B] ∧ [A → A ∈ T] ∧ [B → B ∈ T] A → B: an algo converts a proof of A to a proof of B A ⇛ B: A ∧ [A ∈ T] → B ∧ [B ∈ T]

¬A: A → (0 = 1) ∼A: A ⇛ (0 = 1)

slide-53
SLIDE 53

Introduction NSA and Constructive Analysis Philosophical implications

Lost in translation

BISH (based on IL) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and transfer (T)

A V B: [A ∨ B] ∧ [A → A ∈ T] ∧ [B → B ∈ T] A → B: an algo converts a proof of A to a proof of B A ⇛ B: A ∧ [A ∈ T] → B ∧ [B ∈ T]

¬A: A → (0 = 1) ∼A: A ⇛ (0 = 1)

slide-54
SLIDE 54

Introduction NSA and Constructive Analysis Philosophical implications

Lost in translation

BISH (based on IL) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and transfer (T)

A V B: [A ∨ B] ∧ [A → A ∈ T] ∧ [B → B ∈ T] A → B: an algo converts a proof of A to a proof of B A ⇛ B: A ∧ [A ∈ T] → B ∧ [B ∈ T]

¬A: A → (0 = 1) ∼A: A ⇛ (0 = 1) ≈ “an algo decides if A or if B”

slide-55
SLIDE 55

Introduction NSA and Constructive Analysis Philosophical implications

Lost in translation

BISH (based on IL) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and transfer (T)

A V B: [A ∨ B] ∧ [A → A ∈ T] ∧ [B → B ∈ T] A → B: an algo converts a proof of A to a proof of B A ⇛ B: A ∧ [A ∈ T] → B ∧ [B ∈ T]

¬A: A → (0 = 1) ∼A: A ⇛ (0 = 1) ≈ “an algo decides if A or if B” ≈ “an Ω-inv.proc. decides if A or if B”

slide-56
SLIDE 56

Introduction NSA and Constructive Analysis Philosophical implications

Lost in translation

BISH (based on IL) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and transfer (T)

A V B: [A ∨ B] ∧ [A → A ∈ T] ∧ [B → B ∈ T] A → B: an algo converts a proof of A to a proof of B A ⇛ B: A ∧ [A ∈ T] → B ∧ [B ∈ T]

¬A: A → (0 = 1) ∼A: A ⇛ (0 = 1) ≈ “an algo decides if A or if B” ≈ “an Ω-inv.proc. decides if A or if B”

(∃x)A(x): an algo computes x0 such that A(x0)

slide-57
SLIDE 57

Introduction NSA and Constructive Analysis Philosophical implications

Lost in translation

BISH (based on IL) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and transfer (T)

A V B: [A ∨ B] ∧ [A → A ∈ T] ∧ [B → B ∈ T] A → B: an algo converts a proof of A to a proof of B A ⇛ B: A ∧ [A ∈ T] → B ∧ [B ∈ T]

¬A: A → (0 = 1) ∼A: A ⇛ (0 = 1) ≈ “an algo decides if A or if B” ≈ “an Ω-inv.proc. decides if A or if B”

(∃x)A(x): an algo computes x0 such that A(x0) (∃x)A(x): an Ω-inv. proc. computes x0 such that A(x0)

slide-58
SLIDE 58

Introduction NSA and Constructive Analysis Philosophical implications

Lost in translation

BISH (based on IL) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and transfer (T)

A V B: [A ∨ B] ∧ [A → A ∈ T] ∧ [B → B ∈ T] A → B: an algo converts a proof of A to a proof of B A ⇛ B: A ∧ [A ∈ T] → B ∧ [B ∈ T]

¬A: A → (0 = 1) ∼A: A ⇛ (0 = 1) ≈ “an algo decides if A or if B” ≈ “an Ω-inv.proc. decides if A or if B”

(∃x)A(x): an algo computes x0 such that A(x0) (∃x)A(x): an Ω-inv. proc. computes x0 such that A(x0)

∼[(∀n ∈ N)A(n)]

slide-59
SLIDE 59

Introduction NSA and Constructive Analysis Philosophical implications

Lost in translation

BISH (based on IL) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and transfer (T)

A V B: [A ∨ B] ∧ [A → A ∈ T] ∧ [B → B ∈ T] A → B: an algo converts a proof of A to a proof of B A ⇛ B: A ∧ [A ∈ T] → B ∧ [B ∈ T]

¬A: A → (0 = 1) ∼A: A ⇛ (0 = 1) ≈ “an algo decides if A or if B” ≈ “an Ω-inv.proc. decides if A or if B”

(∃x)A(x): an algo computes x0 such that A(x0) (∃x)A(x): an Ω-inv. proc. computes x0 such that A(x0)

∼[(∀n ∈ N)A(n)] ≡ (∃n ∈ N1)¬A(n) WEAKER than (∃n ∈ N)¬A(n).

slide-60
SLIDE 60

Introduction NSA and Constructive Analysis Philosophical implications

Lost in translation

BISH (based on IL) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and transfer (T)

A V B: [A ∨ B] ∧ [A → A ∈ T] ∧ [B → B ∈ T] A → B: an algo converts a proof of A to a proof of B A ⇛ B: A ∧ [A ∈ T] → B ∧ [B ∈ T]

¬A: A → (0 = 1) ∼A: A ⇛ (0 = 1) ≈ “an algo decides if A or if B” ≈ “an Ω-inv.proc. decides if A or if B”

(∃x)A(x): an algo computes x0 such that A(x0) (∃x)A(x): an Ω-inv. proc. computes x0 such that A(x0)

∼[(∀n ∈ N)A(n)] ≡ (∃n ∈ N1)¬A(n) WEAKER than (∃n ∈ N)¬A(n). ¬[(∀n ∈ N)A(n)] is WEAKER than (∃n ∈ N)¬A(n).

slide-61
SLIDE 61

Introduction NSA and Constructive Analysis Philosophical implications

Lost in translation

BISH (based on IL) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and transfer (T)

A V B: [A ∨ B] ∧ [A → A ∈ T] ∧ [B → B ∈ T] A → B: an algo converts a proof of A to a proof of B A ⇛ B: A ∧ [A ∈ T] → B ∧ [B ∈ T]

¬A: A → (0 = 1) ∼A: A ⇛ (0 = 1) ≈ “an algo decides if A or if B” ≈ “an Ω-inv.proc. decides if A or if B”

(∃x)A(x): an algo computes x0 such that A(x0) (∃x)A(x): an Ω-inv. proc. computes x0 such that A(x0)

WHY is this a good/faithful/reasonable/. . . translation?

slide-62
SLIDE 62

Introduction NSA and Constructive Analysis Philosophical implications

Lost in translation

BISH (based on IL) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and transfer (T)

A V B: [A ∨ B] ∧ [A → A ∈ T] ∧ [B → B ∈ T] A → B: an algo converts a proof of A to a proof of B A ⇛ B: A ∧ [A ∈ T] → B ∧ [B ∈ T]

¬A: A → (0 = 1) ∼A: A ⇛ (0 = 1) ≈ “an algo decides if A or if B” ≈ “an Ω-inv.proc. decides if A or if B”

(∃x)A(x): an algo computes x0 such that A(x0) (∃x)A(x): an Ω-inv. proc. computes x0 such that A(x0)

WHY is this a good/faithful/reasonable/. . . translation? BECAUSE the non-algorithmic/non-constructive principles behave the same!

slide-63
SLIDE 63

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics

slide-64
SLIDE 64

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic

slide-65
SLIDE 65

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ1, P ∨ ¬P

slide-66
SLIDE 66

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ1, P ∨ ¬P

  • LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
slide-67
SLIDE 67

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ1, P ∨ ¬P

  • LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
  • MCT: monotone convergence thm
slide-68
SLIDE 68

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ1, P ∨ ¬P

  • LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
  • MCT: monotone convergence thm
  • CIT: Cantor intersection thm
slide-69
SLIDE 69

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ1, P ∨ ¬P

  • LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
  • MCT: monotone convergence thm
  • CIT: Cantor intersection thm

non-Ω-invariant

slide-70
SLIDE 70

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ1, P ∨ ¬P

  • LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
  • MCT: monotone convergence thm
  • CIT: Cantor intersection thm

non-Ω-invariant LPO: For P ∈ Σ1, P V ∼P

slide-71
SLIDE 71

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ1, P ∨ ¬P

  • LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
  • MCT: monotone convergence thm
  • CIT: Cantor intersection thm

non-Ω-invariant LPO: For P ∈ Σ1, P V ∼P

  • LPR: (∀x ∈ R)(x > 0 V ∼(x < 0))
slide-72
SLIDE 72

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ1, P ∨ ¬P

  • LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
  • MCT: monotone convergence thm
  • CIT: Cantor intersection thm

non-Ω-invariant LPO: For P ∈ Σ1, P V ∼P

  • LPR: (∀x ∈ R)(x > 0 V ∼(x < 0))
  • MCT: monotone convergence thm
slide-73
SLIDE 73

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ1, P ∨ ¬P

  • LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
  • MCT: monotone convergence thm
  • CIT: Cantor intersection thm

non-Ω-invariant LPO: For P ∈ Σ1, P V ∼P

  • LPR: (∀x ∈ R)(x > 0 V ∼(x < 0))
  • MCT: monotone convergence thm
  • CIT: Cantor intersection thm
slide-74
SLIDE 74

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ1, P ∨ ¬P

  • LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
  • MCT: monotone convergence thm
  • CIT: Cantor intersection thm

non-Ω-invariant LPO: For P ∈ Σ1, P V ∼P

  • LPR: (∀x ∈ R)(x > 0 V ∼(x < 0))
  • MCT: monotone convergence thm
  • CIT: Cantor intersection thm

(limit computed by algo)

slide-75
SLIDE 75

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ1, P ∨ ¬P

  • LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
  • MCT: monotone convergence thm
  • CIT: Cantor intersection thm

non-Ω-invariant LPO: For P ∈ Σ1, P V ∼P

  • LPR: (∀x ∈ R)(x > 0 V ∼(x < 0))
  • MCT: monotone convergence thm
  • CIT: Cantor intersection thm

(limit computed by algo) (limit computed by Ω-inv. proc.)

slide-76
SLIDE 76

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ1, P ∨ ¬P

  • LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
  • MCT: monotone convergence thm
  • CIT: Cantor intersection thm

non-Ω-invariant LPO: For P ∈ Σ1, P V ∼P

  • LPR: (∀x ∈ R)(x > 0 V ∼(x < 0))
  • MCT: monotone convergence thm
  • CIT: Cantor intersection thm

(limit computed by algo) (limit computed by Ω-inv. proc.) (point in intersection computed by algo)

slide-77
SLIDE 77

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ1, P ∨ ¬P

  • LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
  • MCT: monotone convergence thm
  • CIT: Cantor intersection thm

non-Ω-invariant LPO: For P ∈ Σ1, P V ∼P

  • LPR: (∀x ∈ R)(x > 0 V ∼(x < 0))
  • MCT: monotone convergence thm
  • CIT: Cantor intersection thm

(limit computed by algo) (limit computed by Ω-inv. proc.) (point in intersection computed by algo) (point in intersection computed by Ω-inv. proc.)

slide-78
SLIDE 78

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ1, P ∨ ¬P

  • LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
  • MCT: monotone convergence thm
  • CIT: Cantor intersection thm

non-Ω-invariant LPO: For P ∈ Σ1, P V ∼P

  • LPR: (∀x ∈ R)(x > 0 V ∼(x < 0))
  • MCT: monotone convergence thm
  • CIT: Cantor intersection thm

(limit computed by algo) (limit computed by Ω-inv. proc.)

  • Universal Transfer

(∀n ∈ N)ϕ(n) → (∀n ∈ ∗N)ϕ(n)

slide-79
SLIDE 79

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics II

slide-80
SLIDE 80

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics II

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic

slide-81
SLIDE 81

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics II

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LLPO

For P, Q ∈ Σ1, ¬(P ∧ Q) → ¬P ∨ ¬Q

slide-82
SLIDE 82

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics II

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LLPO

For P, Q ∈ Σ1, ¬(P ∧ Q) → ¬P ∨ ¬Q

  • LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
slide-83
SLIDE 83

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics II

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LLPO

For P, Q ∈ Σ1, ¬(P ∧ Q) → ¬P ∨ ¬Q

  • LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
  • NIL

(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)

slide-84
SLIDE 84

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics II

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LLPO

For P, Q ∈ Σ1, ¬(P ∧ Q) → ¬P ∨ ¬Q

  • LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
  • NIL

(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)

  • IVT: Intermediate value theorem
slide-85
SLIDE 85

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics II

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LLPO

For P, Q ∈ Σ1, ¬(P ∧ Q) → ¬P ∨ ¬Q

  • LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
  • NIL

(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)

  • IVT: Intermediate value theorem

non-Ω-invariant

slide-86
SLIDE 86

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics II

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LLPO

For P, Q ∈ Σ1, ¬(P ∧ Q) → ¬P ∨ ¬Q

  • LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
  • NIL

(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)

  • IVT: Intermediate value theorem

non-Ω-invariant LLPO

For P, Q ∈ Σ1, ∼(P ∧ Q) ⇛ ∼P V ∼Q

slide-87
SLIDE 87

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics II

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LLPO

For P, Q ∈ Σ1, ¬(P ∧ Q) → ¬P ∨ ¬Q

  • LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
  • NIL

(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)

  • IVT: Intermediate value theorem

non-Ω-invariant LLPO

For P, Q ∈ Σ1, ∼(P ∧ Q) ⇛ ∼P V ∼Q

  • LLPR: (∀x ∈ R)(x ≥ 0 V x ≤ 0)
slide-88
SLIDE 88

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics II

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LLPO

For P, Q ∈ Σ1, ¬(P ∧ Q) → ¬P ∨ ¬Q

  • LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
  • NIL

(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)

  • IVT: Intermediate value theorem

non-Ω-invariant LLPO

For P, Q ∈ Σ1, ∼(P ∧ Q) ⇛ ∼P V ∼Q

  • LLPR: (∀x ∈ R)(x ≥ 0 V x ≤ 0)
  • NIL

(∀x, y ∈ R)(xy = 0 ⇛ x = 0 V y = 0)

slide-89
SLIDE 89

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics II

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LLPO

For P, Q ∈ Σ1, ¬(P ∧ Q) → ¬P ∨ ¬Q

  • LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
  • NIL

(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)

  • IVT: Intermediate value theorem

non-Ω-invariant LLPO

For P, Q ∈ Σ1, ∼(P ∧ Q) ⇛ ∼P V ∼Q

  • LLPR: (∀x ∈ R)(x ≥ 0 V x ≤ 0)
  • NIL

(∀x, y ∈ R)(xy = 0 ⇛ x = 0 V y = 0)

  • IVT: Intermediate value theorem
slide-90
SLIDE 90

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics II

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LLPO

For P, Q ∈ Σ1, ¬(P ∧ Q) → ¬P ∨ ¬Q

  • LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
  • NIL

(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)

  • IVT: Intermediate value theorem

non-Ω-invariant LLPO

For P, Q ∈ Σ1, ∼(P ∧ Q) ⇛ ∼P V ∼Q

  • LLPR: (∀x ∈ R)(x ≥ 0 V x ≤ 0)
  • NIL

(∀x, y ∈ R)(xy = 0 ⇛ x = 0 V y = 0)

  • IVT: Intermediate value theorem

(int. value computed by algo)

slide-91
SLIDE 91

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics II

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LLPO

For P, Q ∈ Σ1, ¬(P ∧ Q) → ¬P ∨ ¬Q

  • LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
  • NIL

(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)

  • IVT: Intermediate value theorem

non-Ω-invariant LLPO

For P, Q ∈ Σ1, ∼(P ∧ Q) ⇛ ∼P V ∼Q

  • LLPR: (∀x ∈ R)(x ≥ 0 V x ≤ 0)
  • NIL

(∀x, y ∈ R)(xy = 0 ⇛ x = 0 V y = 0)

  • IVT: Intermediate value theorem

(int. value computed by algo) (int. value computed by Ω-inv. proc.)

slide-92
SLIDE 92

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics II

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LLPO

For P, Q ∈ Σ1, ¬(P ∧ Q) → ¬P ∨ ¬Q

  • LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
  • NIL

(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)

  • IVT: Intermediate value theorem

non-Ω-invariant LLPO

For P, Q ∈ Σ1, ∼(P ∧ Q) ⇛ ∼P V ∼Q

  • LLPR: (∀x ∈ R)(x ≥ 0 V x ≤ 0)
  • NIL

(∀x, y ∈ R)(xy = 0 ⇛ x = 0 V y = 0)

  • IVT: Intermediate value theorem

(int. value computed by algo) (int. value computed by Ω-inv. proc.) Axioms of R: ¬(x > 0 ∧ x < 0)

slide-93
SLIDE 93

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics II

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LLPO

For P, Q ∈ Σ1, ¬(P ∧ Q) → ¬P ∨ ¬Q

  • LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
  • NIL

(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)

  • IVT: Intermediate value theorem

non-Ω-invariant LLPO

For P, Q ∈ Σ1, ∼(P ∧ Q) ⇛ ∼P V ∼Q

  • LLPR: (∀x ∈ R)(x ≥ 0 V x ≤ 0)
  • NIL

(∀x, y ∈ R)(xy = 0 ⇛ x = 0 V y = 0)

  • IVT: Intermediate value theorem

(int. value computed by algo) (int. value computed by Ω-inv. proc.) Axioms of R: ¬(x > 0 ∧ x < 0) Axioms of R: ∼(x > 0 ∧ x < 0)

slide-94
SLIDE 94

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics III

slide-95
SLIDE 95

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics III

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic

slide-96
SLIDE 96

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics III

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ1, ¬¬P → P

slide-97
SLIDE 97

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics III

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ1, ¬¬P → P

  • MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0)
slide-98
SLIDE 98

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics III

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ1, ¬¬P → P

  • MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0)
  • EXT: the extensionality theorem
slide-99
SLIDE 99

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics III

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ1, ¬¬P → P

  • MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0)
  • EXT: the extensionality theorem

non-Ω-invariant

slide-100
SLIDE 100

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics III

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ1, ¬¬P → P

  • MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0)
  • EXT: the extensionality theorem

non-Ω-invariant MP: For P ∈ Σ1, ∼∼P ⇛ P

slide-101
SLIDE 101

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics III

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ1, ¬¬P → P

  • MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0)
  • EXT: the extensionality theorem

non-Ω-invariant MP: For P ∈ Σ1, ∼∼P ⇛ P

  • MPR: (∀x ∈ R)(∼∼(x > 0) ⇛ x > 0)
slide-102
SLIDE 102

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics III

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ1, ¬¬P → P

  • MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0)
  • EXT: the extensionality theorem

non-Ω-invariant MP: For P ∈ Σ1, ∼∼P ⇛ P

  • MPR: (∀x ∈ R)(∼∼(x > 0) ⇛ x > 0)
  • EXT: the extensionality theorem
slide-103
SLIDE 103

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics III

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ1, ¬¬P → P

  • MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0)
  • EXT: the extensionality theorem

non-Ω-invariant MP: For P ∈ Σ1, ∼∼P ⇛ P

  • MPR: (∀x ∈ R)(∼∼(x > 0) ⇛ x > 0)
  • EXT: the extensionality theorem

WLPO: For P ∈ Σ1, ¬¬P ∨ ¬P

slide-104
SLIDE 104

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics III

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ1, ¬¬P → P

  • MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0)
  • EXT: the extensionality theorem

non-Ω-invariant MP: For P ∈ Σ1, ∼∼P ⇛ P

  • MPR: (∀x ∈ R)(∼∼(x > 0) ⇛ x > 0)
  • EXT: the extensionality theorem

WLPO: For P ∈ Σ1, ¬¬P ∨ ¬P

  • WLPR: (∀x ∈ R)
  • ¬¬(x > 0) ∨ ¬(x > 0)
slide-105
SLIDE 105

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics III

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ1, ¬¬P → P

  • MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0)
  • EXT: the extensionality theorem

non-Ω-invariant MP: For P ∈ Σ1, ∼∼P ⇛ P

  • MPR: (∀x ∈ R)(∼∼(x > 0) ⇛ x > 0)
  • EXT: the extensionality theorem

WLPO: For P ∈ Σ1, ¬¬P ∨ ¬P

  • WLPR: (∀x ∈ R)
  • ¬¬(x > 0) ∨ ¬(x > 0)
  • DISC:

A discontinuous 2N → N-function exists.

slide-106
SLIDE 106

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics III

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ1, ¬¬P → P

  • MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0)
  • EXT: the extensionality theorem

non-Ω-invariant MP: For P ∈ Σ1, ∼∼P ⇛ P

  • MPR: (∀x ∈ R)(∼∼(x > 0) ⇛ x > 0)
  • EXT: the extensionality theorem

WLPO: For P ∈ Σ1, ¬¬P ∨ ¬P

  • WLPR: (∀x ∈ R)
  • ¬¬(x > 0) ∨ ¬(x > 0)
  • DISC:

A discontinuous 2N → N-function exists. WLPO: For P ∈ Σ1, ∼∼P V ∼P

slide-107
SLIDE 107

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics III

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ1, ¬¬P → P

  • MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0)
  • EXT: the extensionality theorem

non-Ω-invariant MP: For P ∈ Σ1, ∼∼P ⇛ P

  • MPR: (∀x ∈ R)(∼∼(x > 0) ⇛ x > 0)
  • EXT: the extensionality theorem

WLPO: For P ∈ Σ1, ¬¬P ∨ ¬P

  • WLPR: (∀x ∈ R)
  • ¬¬(x > 0) ∨ ¬(x > 0)
  • DISC:

A discontinuous 2N → N-function exists. WLPO: For P ∈ Σ1, ∼∼P V ∼P

  • WLPR: (∀x ∈ R)
  • ∼∼(x > 0) V ∼(x > 0)
slide-108
SLIDE 108

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics III

BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ1, ¬¬P → P

  • MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0)
  • EXT: the extensionality theorem

non-Ω-invariant MP: For P ∈ Σ1, ∼∼P ⇛ P

  • MPR: (∀x ∈ R)(∼∼(x > 0) ⇛ x > 0)
  • EXT: the extensionality theorem

WLPO: For P ∈ Σ1, ¬¬P ∨ ¬P

  • WLPR: (∀x ∈ R)
  • ¬¬(x > 0) ∨ ¬(x > 0)
  • DISC:

A discontinuous 2N → N-function exists. WLPO: For P ∈ Σ1, ∼∼P V ∼P

  • WLPR: (∀x ∈ R)
  • ∼∼(x > 0) V ∼(x > 0)
  • DISC:

A discontinuous 2

∗N → ∗N-function exists.

slide-109
SLIDE 109

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics

slide-110
SLIDE 110

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics

Same for WMP, FAN∆, BD-N, and MP∨.

slide-111
SLIDE 111

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics

Same for WMP, FAN∆, BD-N, and MP∨. Same for ‘mixed’ theorems:

slide-112
SLIDE 112

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics

Same for WMP, FAN∆, BD-N, and MP∨. Same for ‘mixed’ theorems: BISH (based on IL) NSA (based on CL)

slide-113
SLIDE 113

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics

Same for WMP, FAN∆, BD-N, and MP∨. Same for ‘mixed’ theorems: BISH (based on IL) NSA (based on CL) LPO ↔ MP+WLPO

slide-114
SLIDE 114

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics

Same for WMP, FAN∆, BD-N, and MP∨. Same for ‘mixed’ theorems: BISH (based on IL) NSA (based on CL) LPO ↔ MP+WLPO MP ↔ WMP + MP∨

slide-115
SLIDE 115

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics

Same for WMP, FAN∆, BD-N, and MP∨. Same for ‘mixed’ theorems: BISH (based on IL) NSA (based on CL) LPO ↔ MP+WLPO MP ↔ WMP + MP∨ WLPO → LLPO

slide-116
SLIDE 116

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics

Same for WMP, FAN∆, BD-N, and MP∨. Same for ‘mixed’ theorems: BISH (based on IL) NSA (based on CL) LPO ↔ MP+WLPO MP ↔ WMP + MP∨ WLPO → LLPO LLPO → MP∨

slide-117
SLIDE 117

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics

Same for WMP, FAN∆, BD-N, and MP∨. Same for ‘mixed’ theorems: BISH (based on IL) NSA (based on CL) LPO ↔ MP+WLPO MP ↔ WMP + MP∨ WLPO → LLPO LLPO → MP∨ LPO → BD-N

slide-118
SLIDE 118

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics

Same for WMP, FAN∆, BD-N, and MP∨. Same for ‘mixed’ theorems: BISH (based on IL) NSA (based on CL) LPO ↔ MP+WLPO MP ↔ WMP + MP∨ WLPO → LLPO LLPO → MP∨ LPO → BD-N LPO → FAN∆

slide-119
SLIDE 119

Introduction NSA and Constructive Analysis Philosophical implications

Constructive Reverse Mathematics

Same for WMP, FAN∆, BD-N, and MP∨. Same for ‘mixed’ theorems: BISH (based on IL) NSA (based on CL) LPO ↔ MP+WLPO MP ↔ WMP + MP∨ WLPO → LLPO LLPO → MP∨ LPO → BD-N LPO → FAN∆ LPO ↔ MP + WLPO MP ↔ WMP + MP∨ WLPO → LLPO LLPO → MP∨ LPO → BD-N LPO → FAN∆

slide-120
SLIDE 120

Introduction NSA and Constructive Analysis Philosophical implications

Conclusion

slide-121
SLIDE 121

Introduction NSA and Constructive Analysis Philosophical implications

Conclusion

Reverse-engineering Reverse Mathematics (Fuchino-sensei)

slide-122
SLIDE 122

Introduction NSA and Constructive Analysis Philosophical implications

Conclusion

Reverse-engineering Reverse Mathematics (Fuchino-sensei) For Bishop’s notion of algorithm, we conclude: Algorithmic ≈ Ω-invariant because Non-algorithmic ≈ Non-Ω-invariant.

slide-123
SLIDE 123

Introduction NSA and Constructive Analysis Philosophical implications

Why is there a connection?

Algorithmic ≈ Ω-invariant because Non-algorithmic ≈ Non-Ω-invariant.

slide-124
SLIDE 124

Introduction NSA and Constructive Analysis Philosophical implications

Why is there a connection?

Algorithmic ≈ Ω-invariant because Non-algorithmic ≈ Non-Ω-invariant.

  • NSA

0 1 . . . ω N

∗N

slide-125
SLIDE 125

Introduction NSA and Constructive Analysis Philosophical implications

Why is there a connection?

Algorithmic ≈ Ω-invariant because Non-algorithmic ≈ Non-Ω-invariant.

  • NSA

0 1 . . . ω N

∗N

finite

slide-126
SLIDE 126

Introduction NSA and Constructive Analysis Philosophical implications

Why is there a connection?

Algorithmic ≈ Ω-invariant because Non-algorithmic ≈ Non-Ω-invariant.

  • NSA

0 1 . . . ω N

∗N

finite infinite

slide-127
SLIDE 127

Introduction NSA and Constructive Analysis Philosophical implications

Why is there a connection?

Algorithmic ≈ Ω-invariant because Non-algorithmic ≈ Non-Ω-invariant.

  • NSA

0 1 . . . ω N

∗N

finite infinite

F is a finite operation

slide-128
SLIDE 128

Introduction NSA and Constructive Analysis Philosophical implications

Why is there a connection?

Algorithmic ≈ Ω-invariant because Non-algorithmic ≈ Non-Ω-invariant.

  • NSA

0 1 . . . ω N

∗N

finite infinite

F is a finite operation

  • F

X

slide-129
SLIDE 129

Introduction NSA and Constructive Analysis Philosophical implications

Why is there a connection?

Algorithmic ≈ Ω-invariant because Non-algorithmic ≈ Non-Ω-invariant.

  • NSA

0 1 . . . ω N

∗N

finite infinite

F is a finite operation

  • F

X

  • BISH

0 1 . . . N N

slide-130
SLIDE 130

Introduction NSA and Constructive Analysis Philosophical implications

Why is there a connection?

Algorithmic ≈ Ω-invariant because Non-algorithmic ≈ Non-Ω-invariant.

  • NSA

0 1 . . . ω N

∗N

finite infinite

F is a finite operation

  • F

X

  • BISH

0 1 . . . N N constructive

slide-131
SLIDE 131

Introduction NSA and Constructive Analysis Philosophical implications

Why is there a connection?

Algorithmic ≈ Ω-invariant because Non-algorithmic ≈ Non-Ω-invariant.

  • NSA

0 1 . . . ω N

∗N

finite infinite

F is a finite operation

  • F

X

  • BISH

0 1 . . . N N constructive non-constructive

slide-132
SLIDE 132

Introduction NSA and Constructive Analysis Philosophical implications

Why is there a connection?

Algorithmic ≈ Ω-invariant because Non-algorithmic ≈ Non-Ω-invariant.

  • NSA

0 1 . . . ω N

∗N

finite infinite

F is a finite operation

  • F

X

  • BISH

0 1 . . . N N constructive non-constructive Hε0(x0) x0

slide-133
SLIDE 133

Introduction NSA and Constructive Analysis Philosophical implications

Why is there a connection?

Algorithmic ≈ Ω-invariant because Non-algorithmic ≈ Non-Ω-invariant.

  • NSA

0 1 . . . ω N

∗N

finite infinite

F is a finite operation

  • F

X

  • BISH

0 1 . . . N N constructive non-constructive Hε0(x0) x0

C is a constructive operation

slide-134
SLIDE 134

Introduction NSA and Constructive Analysis Philosophical implications

Why is there a connection?

Algorithmic ≈ Ω-invariant because Non-algorithmic ≈ Non-Ω-invariant.

  • NSA

0 1 . . . ω N

∗N

finite infinite

F is a finite operation

  • F

X

  • BISH

0 1 . . . N N constructive non-constructive Hε0(x0) x0

C is a constructive operation

  • C

X

slide-135
SLIDE 135

Introduction NSA and Constructive Analysis Philosophical implications

Future research: Bounded Arithmetic

Is Ω-invariance useful for Bounded Arithmetic?

slide-136
SLIDE 136

Introduction NSA and Constructive Analysis Philosophical implications

Future research: Bounded Arithmetic

Is Ω-invariance useful for Bounded Arithmetic?

1 P := PRA with bound |f (n, x)| ≤ p(|x|, |n|) (polynomial p).

slide-137
SLIDE 137

Introduction NSA and Constructive Analysis Philosophical implications

Future research: Bounded Arithmetic

Is Ω-invariance useful for Bounded Arithmetic?

1 P := PRA with bound |f (n, x)| ≤ p(|x|, |n|) (polynomial p). 2 P = B := PRA without a bound, but with two sorts of

variables ( x, a) with recursion limited to x.

slide-138
SLIDE 138

Introduction NSA and Constructive Analysis Philosophical implications

Future research: Bounded Arithmetic

Is Ω-invariance useful for Bounded Arithmetic?

1 P := PRA with bound |f (n, x)| ≤ p(|x|, |n|) (polynomial p). 2 P = B := PRA without a bound, but with two sorts of

variables ( x, a) with recursion limited to x.

3

x are normal numbers, already constructed;

  • a are safe numbers, ideal elements without a construction.
slide-139
SLIDE 139

Introduction NSA and Constructive Analysis Philosophical implications

Future research: Bounded Arithmetic

Is Ω-invariance useful for Bounded Arithmetic?

1 P := PRA with bound |f (n, x)| ≤ p(|x|, |n|) (polynomial p). 2 P = B := PRA without a bound, but with two sorts of

variables ( x, a) with recursion limited to x.

3

x are normal numbers, already constructed;

  • a are safe numbers, ideal elements without a construction.

4 The same (intuitive) picture applies:

slide-140
SLIDE 140

Introduction NSA and Constructive Analysis Philosophical implications

Future research: Bounded Arithmetic

Is Ω-invariance useful for Bounded Arithmetic?

1 P := PRA with bound |f (n, x)| ≤ p(|x|, |n|) (polynomial p). 2 P = B := PRA without a bound, but with two sorts of

variables ( x, a) with recursion limited to x.

3

x are normal numbers, already constructed;

  • a are safe numbers, ideal elements without a construction.

4 The same (intuitive) picture applies:

  • BA

0 1 . . . D N

slide-141
SLIDE 141

Introduction NSA and Constructive Analysis Philosophical implications

Future research: Bounded Arithmetic

Is Ω-invariance useful for Bounded Arithmetic?

1 P := PRA with bound |f (n, x)| ≤ p(|x|, |n|) (polynomial p). 2 P = B := PRA without a bound, but with two sorts of

variables ( x, a) with recursion limited to x.

3

x are normal numbers, already constructed;

  • a are safe numbers, ideal elements without a construction.

4 The same (intuitive) picture applies:

  • BA

0 1 . . . D N normal ≈ constructed

slide-142
SLIDE 142

Introduction NSA and Constructive Analysis Philosophical implications

Future research: Bounded Arithmetic

Is Ω-invariance useful for Bounded Arithmetic?

1 P := PRA with bound |f (n, x)| ≤ p(|x|, |n|) (polynomial p). 2 P = B := PRA without a bound, but with two sorts of

variables ( x, a) with recursion limited to x.

3

x are normal numbers, already constructed;

  • a are safe numbers, ideal elements without a construction.

4 The same (intuitive) picture applies:

  • BA

0 1 . . . D N normal ≈ constructed safe ≈ non-constructed

slide-143
SLIDE 143

Introduction NSA and Constructive Analysis Philosophical implications

Future research: Bounded Arithmetic

Is Ω-invariance useful for Bounded Arithmetic?

1 P := PRA with bound |f (n, x)| ≤ p(|x|, |n|) (polynomial p). 2 P = B := PRA without a bound, but with two sorts of

variables ( x, a) with recursion limited to x.

3

x are normal numbers, already constructed;

  • a are safe numbers, ideal elements without a construction.

4 The same (intuitive) picture applies:

  • BA

0 1 . . . D N normal ≈ constructed safe ≈ non-constructed a0 = 2x0 x0

slide-144
SLIDE 144

Introduction NSA and Constructive Analysis Philosophical implications

Future research: Bounded Arithmetic

Is Ω-invariance useful for Bounded Arithmetic?

1 P := PRA with bound |f (n, x)| ≤ p(|x|, |n|) (polynomial p). 2 P = B := PRA without a bound, but with two sorts of

variables ( x, a) with recursion limited to x.

3

x are normal numbers, already constructed;

  • a are safe numbers, ideal elements without a construction.

4 The same (intuitive) picture applies:

  • BA

0 1 . . . D N normal ≈ constructed safe ≈ non-constructed a0 = 2x0 x0

P is a polytime operation

slide-145
SLIDE 145

Introduction NSA and Constructive Analysis Philosophical implications

Future research: Bounded Arithmetic

Is Ω-invariance useful for Bounded Arithmetic?

1 P := PRA with bound |f (n, x)| ≤ p(|x|, |n|) (polynomial p). 2 P = B := PRA without a bound, but with two sorts of

variables ( x, a) with recursion limited to x.

3

x are normal numbers, already constructed;

  • a are safe numbers, ideal elements without a construction.

4 The same (intuitive) picture applies:

  • BA

0 1 . . . D N normal ≈ constructed safe ≈ non-constructed a0 = 2x0 x0

P is a polytime operation

  • P

X

slide-146
SLIDE 146

Introduction NSA and Constructive Analysis Philosophical implications

Future research: Bounded Arithmetic

Is Ω-invariance useful for Bounded Arithmetic?

1 P := PRA with bound |f (n, x)| ≤ p(|x|, |n|) (polynomial p). 2 P = B := PRA without a bound, but with two sorts of

variables ( x, a) with recursion limited to x.

3

x are normal numbers, already constructed;

  • a are safe numbers, ideal elements without a construction.

4 The same (intuitive) picture applies:

  • BA

0 1 . . . D N normal ≈ constructed safe ≈ non-constructed a0 = 2x0 x0

P is a polytime operation

  • P

X P

?

≈ Safe-invariance: (∀ x ∈ D)(∀ a, b ∈ D)

  • f (

x; a) = f ( x; b)

  • .
slide-147
SLIDE 147

Introduction NSA and Constructive Analysis Philosophical implications

Future work: Type Theory

Martin-L¨

  • f intended his type theory as a foundation for BISH.
slide-148
SLIDE 148

Introduction NSA and Constructive Analysis Philosophical implications

Future work: Type Theory

Martin-L¨

  • f intended his type theory as a foundation for BISH.

Can Ω-invariance help capture e.g. Type Theory?

slide-149
SLIDE 149

Introduction NSA and Constructive Analysis Philosophical implications

Future work: Type Theory

Martin-L¨

  • f intended his type theory as a foundation for BISH.

Can Ω-invariance help capture e.g. Type Theory? Homotopy: f (x) g(x)

slide-150
SLIDE 150

Introduction NSA and Constructive Analysis Philosophical implications

Future work: Type Theory

Martin-L¨

  • f intended his type theory as a foundation for BISH.

Can Ω-invariance help capture e.g. Type Theory? Homotopy: f (x) g(x)

  • continuous transformation ht of f to g (t ∈ [0, 1]).
slide-151
SLIDE 151

Introduction NSA and Constructive Analysis Philosophical implications

Future work: Type Theory

Martin-L¨

  • f intended his type theory as a foundation for BISH.

Can Ω-invariance help capture e.g. Type Theory? Homotopy: f (x) g(x)

  • continuous transformation ht of f to g (t ∈ [0, 1]).

ht1(x) ht2(x) ht3(x) . . . . . . . . .

  • h1(x) =

h0(x) =

slide-152
SLIDE 152

Introduction NSA and Constructive Analysis Philosophical implications

Future work: Type Theory

Martin-L¨

  • f intended his type theory as a foundation for BISH.

Can Ω-invariance help capture e.g. Type Theory? Homotopy: f (x) g(x)

  • ✑✑

✑

  • ✏✏✏✏

✏

  • PPPP

P ❅ ❅ ❅ ❅ ❅

mω(x) ≈ kω(x) ≈

  • ❅

❅

  • ◗◗

◗

  • PPPP

P

  • PPPP

P✚✚✚✚✚

slide-153
SLIDE 153

Introduction NSA and Constructive Analysis Philosophical implications

Future work: Type Theory

Martin-L¨

  • f intended his type theory as a foundation for BISH.

Can Ω-invariance help capture e.g. Type Theory? Homotopy: f (x) g(x)

  • ✑✑

✑

  • ✏✏✏✏

✏

  • PPPP

P ❅ ❅ ❅ ❅ ❅

mω(x) ≈ kω(x) ≈

  • ❅

❅

  • ◗◗

◗

  • PPPP

P

  • PPPP

P✚✚✚✚✚

  • PPPP

P

  • ✏✏✏✏

✏

  • ⇓

increment is multiple of 1

ω

slide-154
SLIDE 154

Introduction NSA and Constructive Analysis Philosophical implications

Future work: Type Theory

Martin-L¨

  • f intended his type theory as a foundation for BISH.

Can Ω-invariance help capture e.g. Type Theory? Homotopy: f (x) g(x)

  • ✑✑

✑

  • ✏✏✏✏

✏

  • PPPP

P ❅ ❅ ❅ ❅ ❅

mω(x) ≈ kω(x) ≈

  • ❅

❅

  • ◗◗

◗

  • PPPP

P

  • PPPP

P✚✚✚✚✚

  • PPPP

P

  • ✏✏✏✏

✏

  • ⇓

increment is multiple of 1

ω

ONE basic step

slide-155
SLIDE 155

Introduction NSA and Constructive Analysis Philosophical implications

Future work: Type Theory

Martin-L¨

  • f intended his type theory as a foundation for BISH.

Can Ω-invariance help capture e.g. Type Theory? Homotopy: f (x) g(x)

  • ✑✑

✑

  • ✏✏✏✏

✏

  • PPPP

P ❅ ❅ ❅ ❅ ❅

mω(x) ≈ kω(x) ≈

  • ❅

❅

  • ◗◗

◗

  • PPPP

P

  • PPPP

P✚✚✚✚✚

  • PPPP

P

  • ✏✏✏✏

✏

  • ⇓

ONE basic step . . . ω basic steps . . .

slide-156
SLIDE 156

Introduction NSA and Constructive Analysis Philosophical implications

Future work: Type Theory

Martin-L¨

  • f intended his type theory as a foundation for BISH.

Can Ω-invariance help capture e.g. Type Theory? Homotopy: f (x) g(x)

  • ✑✑

✑

  • ✏✏✏✏

✏

  • PPPP

P ❅ ❅ ❅ ❅ ❅

mω(x) ≈ kω(x) ≈

  • ❅

❅

  • ◗◗

◗

  • PPPP

P

  • PPPP

P✚✚✚✚✚

  • PPPP

P

  • ✏✏✏✏

✏

  • ⇓

ONE basic step . . . ω basic steps . . . Independent of the choice of ω

slide-157
SLIDE 157

Introduction NSA and Constructive Analysis Philosophical implications

Future work: Type Theory

Martin-L¨

  • f intended his type theory as a foundation for BISH.

Can Ω-invariance help capture e.g. Type Theory? Homotopy: f (x) g(x)

  • ✑✑

✑

  • ✏✏✏✏

✏

  • PPPP

P ❅ ❅ ❅ ❅ ❅

mω(x) ≈ kω(x) ≈

  • ❅

❅

  • ◗◗

◗

  • PPPP

P

  • PPPP

P✚✚✚✚✚

  • PPPP

P

  • ✏✏✏✏

✏

  • ⇓

ONE basic step . . . ω basic steps . . . Independent of the choice of ω ≈ Ω-invariant broken-line transformation hω,t of f to g.

slide-158
SLIDE 158

Introduction NSA and Constructive Analysis Philosophical implications

On (anti)realism

In the right framework, Ω-invariance captures Bishop’s notion algorithm indirectly and from the outside.

slide-159
SLIDE 159

Introduction NSA and Constructive Analysis Philosophical implications

On (anti)realism

In the right framework, Ω-invariance captures Bishop’s notion algorithm indirectly and from the outside. An analogy:

slide-160
SLIDE 160

Introduction NSA and Constructive Analysis Philosophical implications

On (anti)realism

In the right framework, Ω-invariance captures Bishop’s notion algorithm indirectly and from the outside. An analogy: Brains in vats VERSUS

slide-161
SLIDE 161

Introduction NSA and Constructive Analysis Philosophical implications

On (anti)realism

In the right framework, Ω-invariance captures Bishop’s notion algorithm indirectly and from the outside. An analogy: Brains in vats VERSUS Intuitionists/constructivists in NSA.

slide-162
SLIDE 162

Introduction NSA and Constructive Analysis Philosophical implications

On (anti)realism

In the right framework, Ω-invariance captures Bishop’s notion algorithm indirectly and from the outside. An analogy: Brains in vats VERSUS Intuitionists/constructivists in NSA. How does X know he is not actually sitting in Y?

slide-163
SLIDE 163

Introduction NSA and Constructive Analysis Philosophical implications

Philosophy of Physics

slide-164
SLIDE 164

Introduction NSA and Constructive Analysis Philosophical implications

Philosophy of Physics

Why is Mathematics in Physics so constructive/computable?

slide-165
SLIDE 165

Introduction NSA and Constructive Analysis Philosophical implications

Philosophy of Physics

Why is Mathematics in Physics so constructive/computable? Indeed, in Physics, calculations are explicit and existence statements come with a construction (symbolically or numerically).

slide-166
SLIDE 166

Introduction NSA and Constructive Analysis Philosophical implications

Philosophy of Physics

Why is Mathematics in Physics so constructive/computable? Indeed, in Physics, calculations are explicit and existence statements come with a construction (symbolically or numerically). The answer: in Physics, an informal version of NSA is used to date. (The notorious ‘epsilon-delta’ method was never adopted).

slide-167
SLIDE 167

Introduction NSA and Constructive Analysis Philosophical implications

Philosophy of Physics

Why is Mathematics in Physics so constructive/computable? Indeed, in Physics, calculations are explicit and existence statements come with a construction (symbolically or numerically). The answer: in Physics, an informal version of NSA is used to date. (The notorious ‘epsilon-delta’ method was never adopted). Also, in Physics, the end result of a calculation should have physical meaning (modeling of reality).

slide-168
SLIDE 168

Introduction NSA and Constructive Analysis Philosophical implications

Philosophy of Physics

Why is Mathematics in Physics so constructive/computable? Indeed, in Physics, calculations are explicit and existence statements come with a construction (symbolically or numerically). The answer: in Physics, an informal version of NSA is used to date. (The notorious ‘epsilon-delta’ method was never adopted). Also, in Physics, the end result of a calculation should have physical meaning (modeling of reality). An end result with physical meaning will not depend on the infinite number/infinitesimal used, i.e. it is Ω-invariant. (Connes)

slide-169
SLIDE 169

Introduction NSA and Constructive Analysis Philosophical implications

On Robustness

Robustness = invariance under variation of parameters.

slide-170
SLIDE 170

Introduction NSA and Constructive Analysis Philosophical implications

On Robustness

Robustness = invariance under variation of parameters. Parts of Reverse Mathematics and Computability are robust.

slide-171
SLIDE 171

Introduction NSA and Constructive Analysis Philosophical implications

On Robustness

Robustness = invariance under variation of parameters. Parts of Reverse Mathematics and Computability are robust. Robustness is a mixture of syntax and semantics.

slide-172
SLIDE 172

Introduction NSA and Constructive Analysis Philosophical implications

On Robustness

Robustness = invariance under variation of parameters. Parts of Reverse Mathematics and Computability are robust. Robustness is a mixture of syntax and semantics.(Ontology of mathematics?)

slide-173
SLIDE 173

Introduction NSA and Constructive Analysis Philosophical implications

On Robustness

Robustness = invariance under variation of parameters. Parts of Reverse Mathematics and Computability are robust. Robustness is a mixture of syntax and semantics.(Ontology of mathematics?) Robustness (for scientific models) is a no-false-positives guarantee.

slide-174
SLIDE 174

Introduction NSA and Constructive Analysis Philosophical implications

On Robustness

Robustness = invariance under variation of parameters. Parts of Reverse Mathematics and Computability are robust. Robustness is a mixture of syntax and semantics.(Ontology of mathematics?) Robustness (for scientific models) is a no-false-positives guarantee. A distant dream: To provide a framework for building scientific models that come with a proof of robustness,

slide-175
SLIDE 175

Introduction NSA and Constructive Analysis Philosophical implications

On Robustness

Robustness = invariance under variation of parameters. Parts of Reverse Mathematics and Computability are robust. Robustness is a mixture of syntax and semantics.(Ontology of mathematics?) Robustness (for scientific models) is a no-false-positives guarantee. A distant dream: To provide a framework for building scientific models that come with a proof of robustness, in the same way as Type Theory provides a framework of building computer programs that come with a proof of correctness.

slide-176
SLIDE 176

Introduction NSA and Constructive Analysis Philosophical implications

Final Thoughts

slide-177
SLIDE 177

Introduction NSA and Constructive Analysis Philosophical implications

Final Thoughts

And what are these [infinitesimals]? [. . . ] They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?

George Berkeley, The Analyst

slide-178
SLIDE 178

Introduction NSA and Constructive Analysis Philosophical implications

Final Thoughts

And what are these [infinitesimals]? [. . . ] They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?

George Berkeley, The Analyst ...there are good reasons to believe that Nonstandard Analysis, in some version or other, will be the analysis of the future. Kurt G¨

  • del
slide-179
SLIDE 179

Introduction NSA and Constructive Analysis Philosophical implications

Final Thoughts

And what are these [infinitesimals]? [. . . ] They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?

George Berkeley, The Analyst ...there are good reasons to believe that Nonstandard Analysis, in some version or other, will be the analysis of the future. Kurt G¨

  • del

We thank the John Templeton Foundation for its generous support!

slide-180
SLIDE 180

Introduction NSA and Constructive Analysis Philosophical implications

Final Thoughts

And what are these [infinitesimals]? [. . . ] They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?

George Berkeley, The Analyst ...there are good reasons to believe that Nonstandard Analysis, in some version or other, will be the analysis of the future. Kurt G¨

  • del

We thank the John Templeton Foundation for its generous support!

Thank you for your attention!

slide-181
SLIDE 181

Introduction NSA and Constructive Analysis Philosophical implications

Final Thoughts

And what are these [infinitesimals]? [. . . ] They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?

George Berkeley, The Analyst ...there are good reasons to believe that Nonstandard Analysis, in some version or other, will be the analysis of the future. Kurt G¨

  • del

We thank the John Templeton Foundation for its generous support!

Thank you for your attention!

Any questions?