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#induction

1 post1 participant0 posts today
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Paysages Mathématiques

"[Mathematical induction] is a tool that allows us to move from the finite to the infinite. This tool is always useful because it allows us to skip as many steps as we want, sparing us long, tedious, and monotonous verifications that would quickly become impractical. However, it becomes indispensable as soon as we aim at the general theorem, whose analytical verification would bring us closer and closer [...]" – Henri Poincaré (1854-1912)
#quote #mathematics #math #maths #induction

Photographic portrait of Henri Poincaré, and a quote : "[Mathematical induction] is a tool that allows us to move from the finite to the infinite. This tool is always useful because it allows us to skip as many steps as we want, sparing us long, tedious, and monotonous verifications that would quickly become impractical. However, it becomes indispensable as soon as we aim at the general theorem, whose analytical verification would bring us closer and closer without ever allowing us to reach it."
Public
😀🚲

The 11yo is now regularly cooking their own pasta with the #induction hob, and wow is that a lot less for a parent to worry about than the hot electric stovetop or an open gas flame (it's still scalding hot water though, make sure they know how to manage it.) There is even a timer setting so it just turns off if you forget. Also, I highly recommend raising a self-feeding child 😌

Replied in thread
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Akshar Varma

@BernhardWerner My favorite alternative #proof strategy for #induction proofs are #combinatorial (counting) proofs.

I suppose the standard example might be the proof of the coefficients in the binomial theorem expansion, or for the sum of binomial coefficients being powers of 2. These can be proved by induction, of course, but I'm not sure that's common given how easier it is to do a counting proof. It is also much clearer and avoids tedious algebra.

One I like is proving that the sum 1 + 2 + 3 + ⋯ + 𝑛 is 𝑛 + 1 choose 2, the binomial coefficient \(\binom{n+1}{2}\). Bijection proof, counts the same thing in two ways. The thing being counted is the number of ways of choosing two things (distinct, without repetition) from the set {0, 1, ..., 𝑛}. By definition, it is the binomial coefficient we want. The other way to count is to fix the larger number 𝑘, the remaining choices are any of the 𝑘 numbers from 0 to 𝑘 - 1. Thus, across all possible larger numbers, we get the sum from 1 to n.

An alternative alternate proof of the same, slightly more geometric is as follows: arrange dots in a triangle, 1 on row 1, 2 on row 2, and so on up to row n, with n dots. Add a phantom row of n+1 dots below. We want to add up all dots in first n rows: ∑ 𝑖. If you think of all of this as a binary tree/DAG, then every dot has two children (imagine Pascal's triangle). If you pick any two dots in the phantom row, their common ancestor is unique. So counting dots is same as picking two dots in phantom row. Which is the binomial coefficient we want.

Benjamin and Quinn's book on combinatorial proofs is amazing for interpretations of this form (I learned the first proof from it). See also: en.wikipedia.org/wiki/Combinat

en.wikipedia.orgCombinatorial proof - Wikipedia
Continued thread
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Jon Awbrey

Interpreter and Interpretant • Selection 10.2
• inquiryintoinquiry.com/2025/02
• bsky.app/profile/inquiryintoin

In logical terms, the analogy of experience proceeds by inducing a Rule about the validity of a current knowledge base and then by deducing a Fact, the applicability of that knowledge base to a current experience.

Step 1 is Inductive, abstracting a Rule from a Case and a Fact.

• Case : E_past ⇒ E_poss, Chosen events fairly sample Collective events.
• Fact : E_past ⇒ K_pres, Chosen events support the Knowledge regime.
────────────────────────────────────────────────
• Rule : E_poss ⇒ K_pres, Collective events support the Knowledge regime.

Step 2 is Deductive, admitting a Case to a Rule and arriving at a Fact.

• Case : E_pres ⇒ E_poss, Current events fairly sample Collective events.
• Rule : E_poss ⇒ K_pres, Collective events support the Knowledge regime.
────────────────────────────────────────────────
• Fact : E_pres ⇒ K_pres, Current events support the Knowledge regime.

References —

Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), 40–52.
• pdcnet.org/inquiryct/content/i
• academia.edu/57812482/Interpre

Dewey, J. (1910), How We Think, D.C. Heath, Boston, MA. Reprinted (1991), Prometheus Books, Buffalo, NY.
• gutenberg.org/files/37423/3742

Resources —

Survey of Abduction, Deduction, Induction, Analogy, Inquiry
• inquiryintoinquiry.com/2024/02

Survey of Semiotics, Semiosis, Sign Relations
• inquiryintoinquiry.com/2024/01

#Peirce #Logic #Semiotics #Semiosis #SignRelations
#JohnDewey #Interpreter #Interpretant #Pragmatism
#Abduction #Deduction #Induction #Analogy #Inquiry

Inquiry Into Inquiry · Interpreter and Interpretant • Selection 10
More from Inquiry Into Inquiry
Public
Jon Awbrey

Interpreter and Interpretant • Selection 10.1
• inquiryintoinquiry.com/2025/02
• bsky.app/profile/inquiryintoin

Transfer —

Returning to the scene of Dewey's “Sign of Rain” example, let's continue examining how the transfer of knowledge through the analogy of experience works in that case.

By way of a recap, we began by considering a fragment K_pres of the reasoner's knowledge base which is logically equivalent to a conjunction of two rules.

• K_pres ⇔ (B ⇒ A) ∧ (B ⇒ D).

K_pres may be thought of as a piece of knowledge or item of information allowing for the possibility of certain conditions, expressed in the form of a logical constraint on the present universe of discourse.

Next we found it convenient to express all logical statements in terms of their models, that is, in terms of the primitive circumstances or elements of experience over which they hold true.

• Let E_past be the chosen set of experiences, or the circumstances in mind under “past experience”.

• Let E_poss be the collective set of experiences, or the prospective total of possible circumstances.

• Let E_pres be the current experience, or the circumstances immediately present to the reasoner.

If we think of the knowledge base K_pres as referring to the “regime of experience” over which it is valid, then the sets of models involved in the analogy may be ordered according to the relationships of set inclusion or logical implication existing among them.

Figure 4 shows the subsumption relations involved in the analogy of experience.

Figure 4. Analogy of Experience
• inquiryintoinquiry.files.wordp

#Peirce #Logic #Semiotics #Semiosis #SignRelations
#JohnDewey #Interpreter #Interpretant #Pragmatism
#Abduction #Deduction #Induction #Analogy #Inquiry

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Susan Kaye Quinn 🌱(she/her)

I shot this video w/Penn Environment last fall and the crew did an amazing job putting these clean energy stories together.

*Especially* given the ongoing, escalating assault on the environment, I'm proud that the things I've already done (#solar #induction #EV #native #plants) will keep on making a difference for decades to come, regardless.

"they’re investing not only in cleaner energy for themselves, but also a cleaner world for their grandchildren’s generation."

environmentamerica.org/pennsyl

Stories from our Pittsburgh Solar Tour Meet PennEnvironment volunteers Dave Blair and Susan Quinn and Pastor Dave Carver who are all experiencing the benefits of going solar! environmentamerica.org
Public
🐜🦅

My long-in-preparation Cambridge Element ‘Probability and Inductive Logic’ is now available.

doi.org/10.1017/9781009210171

Abstract: Reasoning from inconclusive evidence, or 'induction', is central to science and any applications we make of it. For that reason alone it demands the attention of philosophers of science. This element explores the prospects of using probability theory to provide an inductive logic: a framework for representing evidential support. Constraints on the ideal evaluation of hypotheses suggest that the overall standing of a hypothesis is represented by its probability in light of the total evidence, and incremental support, or confirmation, indicated by the hypothesis having a higher probability conditional on some evidence than it does unconditionally. This proposal is shown to have the capacity to reconstruct many canons of the scientific method and inductive inference. Along the way, significant objections are discussed, such as the challenge of inductive scepticism, and the objection that the probabilistic approach makes evidential support arbitrary.

Cambridge CoreProbability and Inductive LogicCambridge Core - Logic - Probability and Inductive Logic
#probability#induction#bayesianism
Continued thread
Public *
Jon Awbrey

Interpreter and Interpretant • Selection 9.2
• inquiryintoinquiry.com/2025/01

Transfer —

In logical terms, the analogy of experience proceeds by inducing a Rule about the validity of a current knowledge base and then by deducing a Fact, the applicability of that knowledge base to a current experience.

Step 1 is Inductive, abstracting a Rule from a Case and a Fact.

• Case : E_past ⇒ E_poss, Chosen events fairly sample Collective events.
• Fact : E_past ⇒ K_pres, Chosen events support the Knowledge regime.
────────────────────────────────────────────────
• Rule : E_poss ⇒ K_pres, Collective events support the Knowledge regime.

Step 2 is Deductive, admitting a Case to a Rule and arriving at a Fact.

• Case : E_pres ⇒ E_poss, Current events fairly sample Collective events.
• Rule : E_poss ⇒ K_pres, Collective events support the Knowledge regime.
────────────────────────────────────────────────
• Fact : E_pres ⇒ K_pres, Current events support the Knowledge regime.

References —

Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), 40–52.
• pdcnet.org/inquiryct/content/i
• academia.edu/57812482/Interpre

Dewey, J. (1910), How We Think, D.C. Heath, Boston, MA. Reprinted (1991), Prometheus Books, Buffalo, NY.
• gutenberg.org/files/37423/3742

Resources —

Survey of Abduction, Deduction, Induction, Analogy, Inquiry
• inquiryintoinquiry.com/2024/02

Survey of Semiotics, Semiosis, Sign Relations
• inquiryintoinquiry.com/2024/01

#Peirce #Logic #Semiotics #Semiosis #SignRelations
#JohnDewey #Interpreter #Interpretant #Pragmatism
#Abduction #Deduction #Induction #Analogy #Inquiry

Inquiry Into Inquiry · Interpreter and Interpretant • Selection 9
More from Inquiry Into Inquiry
Public *
Jon Awbrey

Interpreter and Interpretant • Selection 9.1
• inquiryintoinquiry.com/2025/01

Transfer —

Let's examine how the transfer of knowledge through the analogy of experience works in the case of Dewey's “Sign of Rain” example.

For concreteness, consider a fragment K_pres of the reasoner's knowledge base which is logically equivalent to a conjunction of two rules.

• K_pres ⇔ (B ⇒ A) ∧ (B ⇒ D).

K_pres may be thought of as a piece of knowledge or item of information allowing for the possibility of certain conditions, expressed in the form of a logical constraint on the present universe of discourse.

It is convenient to have the option of expressing all logical statements in terms of their models, that is, in terms of the primitive circumstances or elements of experience over which they hold true.

• Let E_past be the chosen set of experiences, or the circumstances in mind under “past experience”.

• Let E_poss be the collective set of experiences, or the prospective total of possible circumstances.

• Let E_pres be the current experience, or the circumstances immediately present to the reasoner.

If we think of the knowledge base K_pres as referring to the “regime of experience” over which it is valid, then the sets of models involved in the analogy may be ordered according to the relationships of set inclusion or logical implication existing among them.

#Peirce #Logic #Semiotics #Semiosis #SignRelations
#JohnDewey #Interpreter #Interpretant #Pragmatism
#Abduction #Deduction #Induction #Analogy #Inquiry

Inquiry Into Inquiry · Interpreter and Interpretant • Selection 9
More from Inquiry Into Inquiry
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