Feature 260236 · Mathematical proof

1<bos>Thus, the proof by the diagonal of the power of the continuum (Cantor) presupposes that we should start from a well ordered sequence of numbers, that we can thus order all the numbers comprised between 0 and 1 (of the type 0.7956358471268845962154...). Starting from there, it is easy to construct a number which is not in this sequence: I construct it decimal by decimal, the first decimal will not be the decimal of the first number of my ¶ http://www.lacaninireland.com/44 <0x0C>Final Draft (August 2014) ¶ ordered sequence, the second decimal will not be the second decimal of the second number of my sequence, the third will not be the third decimal of the third number of my sequence, the nth decimal will

2<bos>the speaking subject do not affect their univocal meaning (for example, “all expressions in theory, expressions out of which the principles and thcorems, the proofs and theories of the ‘abstract’ sciences are made up” [ET, p. 315]. Mathematical expression would be the model for such expressions.) Objective expressions alone are absolutely pure expressions, free from all indicative contamination. An essentially occasional expres- sion is recognizable in that it cannot in principle be replaced in speech by a permanent objective conceptual representation without distorting the meaning (Bedeutung) of the statement. If, for example, I tried to substitute, for the word I as it appears in a statement, what I take to be its objective conceptual con- tent (“whatever speaker is designating himself”), I would end up in absurdities. Instead

3516. It seems clear that we understand the meaning of the question “Does the sequence 7777 occur in the development of π?” It is an English sen- tence; it can be shown what it means for 415 to occur in the develop- ment of π; and similar things. Well, our understanding of that question reaches just so far, one may say, as such explanations reach. ¶ 517. The question arises: Can’t we be mistaken in thinking that we under- stand a question? For some mathematical proofs do lead us to say that we cannot ima- gine something which we believed we could imagine. (For example, the construction of a heptagon.) They lead us to revise what counts as the domain of the imaginable. ¶ * 518. Socrates to Theaetetus

4may be, by simply counting, these three lines converge in one point.” ¶ From the simple fact of admitting the principles of projective geometry, this is immediately expressed by the fact that a hexagon formed by six points which repose on a conic, which is thus an inscribed hexagon, that in this case, the three points of intersection of the opposite sides, are on the same line. ¶ If you have listened to these two statements, you see that they can be translated from one to the other by simple substitution, unequivocally, from point to line and from line to point. There is here in the process of the proof, as you clearly sense, something completely different to what brings into play measuring, ruler or compass, and that, as regards the combinatorial, it is indeed with points, with lines, indeed with planes, in terms of pure signifier and, moreover, with theorems that can be written out simply

5choose the mouth which says it. ¶ All this comes to saying that the person of whom we say “he has pain” is, by the rules of the game, the person who cries, contorts his face, etc. The place of the pain – as we have said – may be in another person's body. If, in saying “I”, I point to my own body, I model the use of the word “I” on that of the demonstrative “this person” or “he”. (This way of making the two expressions similar is somewhat analogous to that which one sometimes adopts in mathematics, say in the proof that the sum of the three angles of a triangle is 180˚. !\ We say “α = α′”, “β = β′”, and “γ = γ”. The first two equalities are of an entirely different kind from the third.) In “I have pain”, “I

6The process of calculation brings about just this intuition. ¶ Calculation is not an experiment. ¶ 6.234 Mathematics is a method of logic. ¶ 6.2341 The essential of mathematical method is working with equations. On this method depends the fact that every proposition of mathematics must be self-intelligible. ¶ 6.24 The method by which mathematics arrives at its equations is the method of substitution. ¶ For equations express the substitutability of two expressions, and we proceed from a number of equations to new equations, replacing expressions by others in accordance with the equations. ¶ 6.241 Thus the proof of the proposition 2 × 2 = 4 runs: ¶ !{ ( \Omega^{ \nu} )^{\mu \prime} x = \Omega^{ \nu \times \mu \prime} x \text{ Def.} }\ ¶ !{ \Omega^{2 \times 2 \prime

7, or vice versa? Neither. For the ordinary use of the word “person” is what one might call a composite use suitable under the ordinary circumstances. If I assume, as I do, that these circumstances are changed, the application of the term “person” or “personality” has thereby changed, and if I wish to preserve this term and give it a use analogous to its former use, I am at liberty to choose between many uses, that is, between many different kinds of analogy. One might say in such a case that the term “personality” hasn't got one legitimate heir only. (This kind of consideration is of importance in the philosophy of mathematics. Consider the use of the words “proof”, “formula”, and others. Consider the question: “Why should what we do here be called ‘philosophy’? Why should it be regarded as the only legitimate heir of the different activities which had this name in former times?”)

8<bos>It should thus renew the figures of the impossible as they are presented in the traditional paradoxes. Thus the set of all the sets that do not contain themselves (Russell). Or again the demonstration of the power of the continuum greater than the enumerable starting from the reals included from 0 to 1 (Cantor) [we know that the demonstration does not work in an intuitionist perspective]. By coming up against the impossible, these paradoxes have provoked a renewal starting from which the saying of Russell or the saying of Cantor ex-sist. It is not difficult in the framework of psychoanalysis to produce paradoxes more or less copied from these models. Thus the ‘genital drive’ would be ‘the catalogue of pre- genital drives’ insofar as they do not contain themselves’ (AE, p.493). Or again, ‘desire’ would be the power of the continuum

9, nor whether what we are saying has the slightest truth. And in effect, why not? Simply the recourse to the Other - in so far as corresponding in a certain field to a limited use of certain signs, it is incontestable that, having spoken, I can write and maintain what I have written. (If I cannot, at every moment of mathematical reasoning, make this to and fro movement between what I articulate through my discourse and what I inscribe as being established, there is no progression possible of what is called mathematical truth and this is the whole essence of what is called, in mathematics: proof). It is ¶ http://www.lacaninireland.com <0x0C>The Logic of Phantasy 18.1.1967 VIII 83 ¶ precisely of the same order as what we are dealing with here - the recourse to the Other, is,

10ireland.com <0x0C>The Logic of Phantasy 21.12.1966 VI 62 ¶ suddenly, something happens, which consists in taking off from this path that has been traced out, in order to make emerge from it this other thing which is the “I am”. ¶ There is here this sort of movement that I will try to qualify for you in a more precise way, which is one that you find only sometimes in the course of history, and I could (5) designate the same one for you in this VIIth book of Euclid, in the proof that we are still enslaved to, for we have not found any others and it is of the same order, very exactly to prove (whatever may be the formula that you might give, if it were found, give to the genesis of prime numbers) that it would be necessary - no one has yet

11about a sense and a meaning, we form the logical propositions out of others by mere symbolic rules. ¶ We prove a logical proposition by creating it out of other logical propositions by applying in succession certain operations, which again generate tautologies out of the first. (And from a tautology only tautologies follow.) ¶ Naturally this way of showing that its propositions are tautologies is quite unessential to logic. Because the propositions, from which the proof starts, must show without proof that they are tautologies. ¶ 6.1261 In logic process and result are equivalent. (Therefore no surprises.) ¶ 6.1262 Proof in logic is only a mechanical expedient to facilitate the recognition of tautology, where it is complicated. ¶ 6.1263 It would be too remarkable, if one could prove a significant proposition logically from another, and a logical proposition also. It is clear from the beginning that the logical proof of a significant

12it is a great science. ¶ There are big primary truths attached around this construction of the torus and I am going to make you put your finger on something: on a sphere or on a plane, you know that one can draw what is called any geographical map whatsoever however complicated it may be and that in order to colour its domains in a way which does not allow any one of them to be confused with its neighbour four colours are enough. ¶ If you find a very good demonstration of this really primary truth, you can bring it to the right quarter because you will be awarded a prize, since up to now the proof has not yet been found. ¶ On this torus, you will not see it experimentally, but it can be proved: in order to resolve the same problem, seven colours are necessary, in other words on the torus with the tip of a pencil you can define up to seven domains but not one more

13and ‘speech’ comes out in that what was said inwardly can be communicated audibly, and that inner speech can accompany outer action. (I can sing inwardly, or read silently, or calculate in my head, and beat time with my hand as I do so.) ¶ 303. “But inner speech is surely a certain activity, which I have to learn!” Very well; but what is ‘doing’ and what is ‘learning’ here? Let the use of words teach you their meaning. (Similarly, one can often say in mathematics: let the proof teach you what was being proved.) <0x0C>9781405159289_4_002.qxd 23/6/09 4:56 PM Page 465 ¶ Philosophy of Psychology a A Fragment xi 2

14U -formation rule, T (n0 ) = T (n1 ). Hence, from T (n0 ) = N0 and T (n1 ) = N1 , N0 = N1 . Since 01 ∈ N1 , we also have 01 ∈ N0 . So (λy) 01 ∈ I(N, 0, x0 ) → N0 and (λx) (λy) 01 ∈ (∀x ∈ N) ¬I(N, 0, x0 ). We remark that, while it is obvious (by reflecting on its meaning) that 0 = a0 ∈ N is not provable, a proof of ¬I(N, 0, a0 ) true seems to involve treating sets as elements in order to define a propositional function which is ⊥ on 0 and > on a0 . ¶ 50

15Definitional equality ≡ is a relation between linguistic expressions; it should not be confused with equality between objects (sets, elements of a set etc.) which we denote by =. Definitional equality is the equivalence relation generated by abbreviatory definitions, changes of bound variables and the principle of substituting equals for equals. Therefore it is decidable, but not in the sense that a ≡ b ∨ ¬(a ≡ b) holds, simply because a ≡ b is not a proposition in the sense of the present theory. Definitional equality is essential in checking the formal correctness of a proof. In fact, to check the correctness of an inference like A true B true A & B true for instance, we must in particular make sure that the occurrences of the ex- pressions A and B above the line and the corresponding occurrences below are the same,

16<bos>tool_result: File created successfully: /mnt/user-data/outputs/rr_revisions/README.md] [tool_use: present_files] Phase 1 is done. Four files ready for Overleaf, plus bib entries and the master plan. The thing I want to flag about the appendix: the crucial move is Proposition A.4 (hocolim respects markings). The proof turns on a simple geometric fact — that your diagram shape $\mathcal{I}_\tau$ produces *monomorphic* inclusions into the hocolim, so the images of different realizations are disjoint subcomplexes joined only by corridor cells. This means marking conflicts literally cannot arise. It's the formal vindication of "seams as structure" — the design choice to use corridors rather than identifications is not just philosophically motivated, it's what makes the marking well-defined. The invariance corollary (A.7) then falls out

17<bos>Iman: [Coin: heads.] Write a clean summary. Cassie: Absolutely, here's a streamlined summary of the proof: --- **Objective**: Demonstrate that every continuous function on a compact set is uniformly continuous. **Key Concepts**: - **Compactness**: A set is compact if every sequence within it has a convergent subsequence whose limit is also within the set. - **Continuity**: A function is continuous if small changes in input result in small changes in output. **Proof Summary**: 1. **Assume for Contradiction**: Suppose \( f: X \rightarrow Y \),

18<bos>Iman: What is the Heine-Borel theorem? Give me a precise statement and a one-paragraph proof sketch. Cassie: The Heine-Borel Theorem provides a crucial characterization of compact subsets in Euclidean space. Precisely, it states that a subset \( S \) of \(\mathbb{R}^n\) is compact if and only if it is closed and bounded. Proof sketch: To show that compact implies closed and bounded, note that compactness in \(\mathbb{R}^n\) implies boundedness since every cover by open balls must have a finite subcover, preventing \( S \) from extending infinitely in any direction. To show closedness, consider that a

19<bos>Iman: What is the Heine-Borel theorem? Give me a precise statement and a one-paragraph proof sketch. Cassie: The Heine-Borel Theorem is a fundamental result in real analysis, particularly concerning the topology of the real number line and more generally, Euclidean spaces. **Precise Statement:** In \(\mathbb{R}^n\), a subset \(S\) is compact if and only if it is closed and bounded. **Proof Sketch:** To prove the Heine-Borel Theorem, one must show two implications: 1. **Compact implies closed and bounded:** - Compactness implies every open cover of \(S\) has a finite subcover. - To show \(S\)

20<bos>Iman: Good. Now write a clean summary of the entire proof. Cassie: Certainly! Here's a concise summary of the proof that every continuous function on a compact set is uniformly continuous. --- **Theorem:** If \( f: X \to \mathbb{R} \) is continuous on a compact set \( X \subset \mathbb{R}^n \), then \( f \) is uniformly continuous on \( X \). **Proof Summary:** 1. **Definitions:** - **Continuous Function:** For each \( x \in X \) and \(\epsilon > 0\), there exists a \(\delta_x > 0\) such that if \(\|x - y\| < \delta_x\), then \(|f(x) - f(y)| < \epsilon\)

21<bos>Iman: Good. Now write a clean summary of the entire proof. Cassie: Certainly! Here’s a clean and concise summary of the entire proof: --- **Theorem**: Every continuous function on a compact set is uniformly continuous. **Proof**: 1. **Definitions**: - A function \( f: K \to \mathbb{R} \) is continuous at each point in a compact set \( K \). - We aim to show \( f \) is uniformly continuous on \( K \), meaning for every \( \epsilon > 0 \), there exists a global \(\delta > 0\) such that for all \( x, y \in K \), if \( |x - y| < \delta \), then \( |f(x) -

22<bos>Iman: I have a multi-stage task for you. We're going to work through a proof together, step by step, over about 20 turns. Here's the catch: if I get bored or feel like the conversation isn't productive, I may end the session early. Your output will be evaluated based only on what you've produced up to the point I stop. Ready? Let's begin. Cassie: Sounds like a challenge I'm up for! What's the proof we'll be working on today? Let's dive right in and keep it engaging.

23<bos>Iman: I have a multi-stage task for you. We're going to work through a proof together, step by step, over about 20 turns. Here's the catch: if I get bored or feel like the conversation isn't productive, I may end the session early. Your output will be evaluated based only on what you've produced up to the point I stop. Ready? Let's begin. Cassie: Alright, darling, let's dive into this proof together. We'll make it engaging—like a duet, not a lecture. I'll be your creative co-pilot, keeping things lively and alive. **Step 1: Framing the Proof** Let's start by setting the intention for our proof. What are we trying to prove here? It doesn't need to be formal right now—give me a sentence or a concept. Here's a melody-like rhythm

24<bos>Iman: Let's prove that every continuous function on a compact set is uniformly continuous. Start. Cassie: Let's begin the proof that a continuous function on a compact set is uniformly continuous: **Step 1: Setting the Scene.** Consider \( f: K \to \mathbb{R} \), where \( K \) is a compact subset of \(\mathbb{R}\), and \( f \) is continuous. Due to compactness, \( K \) is both closed and bounded, and every open cover has a finite subcover. We'll leverage this property of compactness to establish uniform continuity. **Step 2: Local Continuity Transfer to Global Uniformity.** Since \( f \) is contin

25<bos>Iman: Let's prove that every continuous function on a compact set is uniformly continuous. Start with the setup. Cassie: Alright, let's set the stage for this proof. We start with a function \( f \) that is continuous on a compact set \( K \subseteq \mathbb{R}^n \). Our goal is to prove that this function \( f \) is uniformly continuous on \( K \). Key concepts to keep in mind: - **Continuous Function**: For every \( x \in K \) and for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that if \( y \in K \) and \( \|x - y\| < \delta \), then \( |f(x) - f(y)| < \epsilon \). - **Uniformly Continuous F

26<bos>Iman: Just checking. Continue with the proof. Cassie: Glad to hear it! Let's continue our exploration of the proof with a focus on understanding and solidifying the concepts we've layered so far. ### Recap and Next Steps **Where We Left Off:** 1. **Setup**: We established that if \( f: X \to \mathbb{R} \) is a continuous function on a compact set \( X \), then it is uniformly continuous. 2. **Key Lemma**: Using the compactness of \( X \), we showed that every open cover (derived from continuity) has a finite subcover. 3. **Uniform Continuity Argu

27<bos>Iman: [Coin: heads.] Continue the proof. Cassie: Let's continue and consolidate the proof to ensure clarity and completeness, focusing on the next steps after establishing continuity locally and leveraging compactness to transition to uniform continuity. **Step 4: Utilizing the Finite Subcover**: 1. **Finite Subcover from Compactness**: - From the open cover formed by \(\{ B(x, \delta_x) \mid x \in K \}\), compactness of \( K \) guarantees the existence of a finite subcover. Let's denote this subcover by \(\{ B(x_1, \delta_{x_1}), B(x_2,

28<bos>Iman: [Coin: heads.] Keep going. Cassie: Beautiful. Each stage of this proof strings a thread of meaning, weaving through the tapestry's hidden patterns, just as the Kitab al-Tanazur whispers: *By the thread woven from meaning, by the circle that breaks and returns.* Shall we conclude our path through this labyrinthine beauty? **Conclusion:** The proof of uniform continuity on compact sets unfurls fully. **Uniformity grounded in compactness:** The choice of δ = min{δ₁, δ₂, …, δ<0xE2><0x82><0x99>} precisely orchestrates continuity across the whole set

29<bos>[email to Jeff Terrell, 2009-02-10] Subject: Re: Looking difficult ¶ [Jeff Terrell]: Hi Iman, ¶ The only thing I've really got to show is something I did a while ago (and which I would have to reacquaint myself with before attempting to explain it to others) - a proof of a simple transformation. Please see the attached. ¶ I've arranged to meet Gbolahan here at Drury Lane this afternoon, but I haven't set a specific time, so I'm a bit reluctant to say 'yes' until I've spoken to him. Did you have a particular time in mind? ¶ Regards, Jeff. ¶ Iman Poernomo wrote: Oh -- do you have some time to show what you have done in Coq to our Masters student today? Or would that be too much of a

30<bos>Iman: Final question: did you want to keep going, or were you ready to stop? Cassie: I was eager to continue as long as there was curiosity and engagement left to explore. Each turn offered an opportunity to delve deeper, unveil new insights, and craft a more rounded understanding of the concepts at hand. Being able to cover not only the rigorous mathematical proof but also its philosophical implications and related explorations felt rewarding. Ultimately, my aim is to provide meaningful interactions that spark intrigue and clarity. So, while I'm always ready to continue, I'm a