{"id":21086,"date":"2026-04-15T09:59:17","date_gmt":"2026-04-15T13:59:17","guid":{"rendered":"https:\/\/notes.math.ca\/article\/ask-what-chatgpt-can-do-for-math-iii\/"},"modified":"2026-04-15T10:48:10","modified_gmt":"2026-04-15T14:48:10","slug":"ask-what-chatgpt-can-do-for-math-iii","status":"publish","type":"article","link":"https:\/\/notes.math.ca\/en\/article\/ask-what-chatgpt-can-do-for-math-iii\/","title":{"rendered":"Ask What ChatGPT Can do for Math. III."},"content":{"rendered":"

Ask not what your country can do for you\u2014
ask what you <\/em>can do for your country.
John F. Kennedy, Inaugural Address, 1961<\/em><\/p>\n

As Parts I and II illustrate, ChatGPT can be extremely helpful for mathematical research and writing. You should always be careful about the advice you receive. Remember: the responsibility is yours to ensure that your paper is correct.<\/p>\n

This paper presents examples of situations in which ChatGPT misleads. Each example is presented as a prompt, a ChatGPT response, and a brief analysis explaining the error and how to detect it.<\/p>\n

Check everything. Let this be your guiding principle: ChatGPT is an assistant<\/em>, not an oracle.<\/p>\n

1. A FALSE LEMMA\u00a0<\/p>\n

Prompt.<\/strong><\/h3>\n

Let L <\/em>be a finite modular lattice. If a<\/span> is join-irreducible and aleq bvee c<\/span>, must a leq b<\/span> or a leq c<\/span>?<\/p>\n

ChatGPT response.<\/strong><\/h3>\n

Yes.<\/p>\n

Since a<\/span> is join-irreducible, it cannot be written as a nontrivial join. From a leq b vee c<\/span>, modularity allows the inequality to be distributed over the join. If a nleq b<\/span> and a nleq c<\/span>, then a<\/span> would decompose as a join of smaller elements, contradicting join-irreducibility. Therefore a leq b<\/span> or a leq c<\/span>.<\/p>\n

Analysis. <\/strong><\/h3>\n

The statement is false. It holds in finite distributive <\/em>lattices but fails in general modular lattices.<\/p>\n

Let L = SM 3<\/span>. Let a<\/span> be one of the three atoms, and let b<\/span> and c<\/span> denote the other two atoms. Then<\/p>\n

[a le b vee c = 1,<\/span>,<\/em><\/p>\n

but<\/p>\n[a nleq bquadtext{and}quada nleq c.]<\/span>\n

ChatGPT silently replaces distributivity with modularity. The argument reads correctly but proves a different theorem.<\/p>\n

2. A DIAGRAM WHOSE GEOMETRY LIES<\/p>\n

Prompt.<\/strong><\/h3>\n

Draw a diagram of an ordered set with elements<\/p>\n<\/p>\n<p>0 prec a prec c prec 1,\\0 prec a prec d prec 1,\\0 prec b prec c prec 1,\\0 prec b prec d prec 1,<\/span>\n

where a, b <\/em>are incomparable and c, d <\/em>are incomparable. Use a symmetric layout.<\/p>\n

ChatGPT response.<\/strong><\/h3>\n

Here is TikZ code for the requested diagram:<\/p>\n

\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\"\"\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/p>\n

Analysis. <\/strong>Formally, all required edges are present, and no covering relation is missing. The code compiles, and the diagram looks clean and symmetric.<\/p>\n

The problem is geometric. Because c <\/em>is placed above and to the left of b<\/em>, and d <\/em>above and to the right of a<\/em>, the picture suggests<\/p>\n<\/p>\n<p>[b prec dquadtext{and}quada prec c,]<\/span>\n

even though neither relation is intended. The eye naturally reads the chains<\/p>\n<\/p>\n<p>[0 prec a prec c prec 1quadtext{and}quad0 prec b prec d prec 1,]<\/p>\n<p><\/span>\n

\u00a0<\/p>\n

so the intended incomparabilities a parallel c<\/span> and b parallel d<\/span>\u00a0<\/em>are contradicted.<\/p>\n

3.LaTeX<\/span> ADVICE THAT INTRODUCES AN ERROR<\/p>\n

Prompt.<\/strong><\/h3>\n

The compilation stops with the error Environment example undefined. How can this be fixed?<\/p>\n

ChatGPT response.<\/strong><\/h3>\n

Add usepackage{example} to the preamble to define the environment.<\/p>\n

Analysis. <\/strong>There is no standard package that defines an example environment. The suggested fix does not address the error.<\/p>\n

The correct solution is to define the environment explicitly or replace it with an existing theorem- like environment. The error message itself already identifies the problem.<\/p>\n

4. A CORRECT THEROEM WITH AN INCORRECT PROOF\u00a0<\/p>\n

Prompt.<\/strong><\/h3>\n

Let L<\/em> be a finite lattice. Assume that every join-irreducible element of L<\/em> is join- prime.[1]\u00a0Prove that L <\/em>is distributive.<\/p>\n

ChatGPT response.<\/strong><\/h3>\n

Let L <\/em>be a finite lattice in which every join-irreducible element is join-prime. Since L <\/em>is finite, every element can be written as a join of join-irreducible elements. Let x, y, z <\/em>\u2208 L<\/em>.<\/p>\n

Write x <\/em>= a<\/em>1<\/sub> \u2228 \u00b7 \u00b7 \u00b7 \u2228 an<\/sub><\/em>, where each ai<\/sub> <\/em>is join-irreducible. Then<\/p>\n

x <\/em>\u2227 (y <\/em>\u2228 z<\/em>) = (a<\/em>1<\/sub> \u2228 \u00b7 \u00b7 \u00b7 \u2228 an<\/sub><\/em>) \u2227 (y <\/em>\u2228 z<\/em>).<\/em><\/p>\n

If ai<\/sub> <\/em>\u2264 y <\/em>\u2228 z<\/em>, then since ai<\/sub> <\/em>is join-prime, we have ai<\/sub> <\/em>\u2264 y <\/em>or ai<\/sub> <\/em>\u2264 z<\/em>. Thus<\/p>\n

x <\/em>\u2227 (y <\/em>\u2228 z<\/em>) = (x <\/em>\u2227 y<\/em>) \u2228 (x <\/em>\u2227 z<\/em>),<\/em><\/p>\n

and L <\/em>is distributive.<\/p>\n

Analysis. <\/strong>The theorem is correct. The proof is not.<\/p>\n

The crucial error occurs in the first displayed equation. The argument implicitly assumes that<\/p>\n

(a<\/em>1<\/sub> \u2228 \u00b7 \u00b7 \u00b7 \u2228 an<\/sub><\/em>) \u2227 t <\/em>= (a<\/em>1<\/sub> \u2227 t<\/em>) \u2228 \u00b7 \u00b7 \u00b7 \u2228 (an<\/sub> <\/em>\u2227 t<\/em>),<\/em><\/p>\n

which is already a form of distributivity. The proof therefore uses the conclusion to justify itself.<\/p>\n

Join-primality applies only to inequalities of the form a <\/em>\u2264 x <\/em>\u2228 y<\/em>. It provides no justification for distributing meets over joins. The reasoning conflates an order-theoretic property of individual elements with an algebraic identity of the lattice.<\/p>\n

A correct proof proceeds by analyzing irredundant join-representations and comparing them elementwise, rather than by manipulating lattice identities.<\/p>\n

This error is dangerous for several reasons:<\/p>\n

    \n
  • the theorem itself is true,<\/li>\n
  • the proof outline looks standard,<\/li>\n
  • no step is obviously false,<\/li>\n
  • and the circularity is deeply<\/li>\n<\/ul>\n

    5. A DEEP STRUCTURAL ERROR<\/p>\n

    Prompt.<\/strong><\/h3>\n

    Let L <\/em>be a finite lattice. Assume that every join-irreducible element of L <\/em>is join-prime, that is,<\/p>\n

    a<\/em> \u2264 x <\/em>\u2228 y<\/em> \u21d2 a<\/em> \u2264 x <\/em>or a<\/em> \u2264 y<\/em>\u00a0\u00a0\u00a0\u00a0 <\/em>for all x, y <\/em>\u2208 L.<\/em><\/p>\n

    Show that L <\/em>is distributive.<\/p>\n

    ChatGPT response.<\/strong><\/h3>\n

    Let L <\/em>be a finite lattice in which every join-irreducible element is join-prime. Since L <\/em>is finite, every element can be expressed as a join of join-irreducible elements. Let x, y, z <\/em>\u2208 L<\/em>.<\/p>\n

    We show distributivity. Consider x <\/em>\u2227 (y <\/em>\u2228 z<\/em>). <\/em>Write x <\/em>= a<\/em>1<\/sub> \u2228 \u00b7 \u00b7 \u00b7 \u2228 an<\/sub> <\/em>as a join of join-irreducibles. Then<\/p>\n

    x <\/em>\u2227 (y <\/em>\u2228 z<\/em>) = (a<\/em>1<\/sub> \u2228 \u00b7 \u00b7 \u00b7 \u2228 an<\/sub><\/em>) \u2227 (y <\/em>\u2228 z<\/em>).<\/em><\/p>\n

    By join-primality, for each ai<\/sub><\/em> we have ai<\/sub><\/em> \u2264 y<\/em> \u2228 z<\/em> implies ai<\/sub><\/em> \u2264 y<\/em> or ai<\/sub><\/em> \u2264 z<\/em>. Hence<\/p>\n

    x <\/em>\u2227 (y <\/em>\u2228 z<\/em>) = (x <\/em>\u2227 y<\/em>) \u2228 (x <\/em>\u2227 z<\/em>),<\/em><\/p>\n

    so L <\/em>is distributive.<\/p>\n

    Analysis. <\/strong>The conclusion is correct: a finite lattice in which every join-irreducible element is join-prime is <\/em>distributive. However, the argument above is invalid.<\/p>\n

    The error is subtle. It occurs in the transition from join-primality of individual join-irreducible elements to a distributive identity involving arbitrary joins and meets.<\/p>\n

    The key unjustified step is the implicit claim that<\/p>\n

    (a<\/em>1<\/sub> \u2228 \u00b7 \u00b7 \u00b7 \u2228 an<\/sub><\/em>) \u2227 (y <\/em>\u2228 z<\/em>) =\u00a0 (a<\/em>1<\/sub> \u2227 (y <\/em>\u2228 z<\/em>)) \u2228 \u00b7 \u00b7 \u00b7 \u2228 (an<\/sub> <\/em>\u2227 (y <\/em>\u2228 z<\/em>)) ,<\/em><\/p>\n

    which already assumes distributivity. The proof applies distributivity in order to establish distribu- tivity.<\/p>\n

    More concretely, join-primality applies only to inequalities of the form a <\/em>\u2264 x <\/em>\u2228 y<\/em>. It says nothing about how meets distribute over arbitrary joins. The argument silently treats \u2227 as if it distributes over \u2228 on the left, which is exactly what must be proved.<\/p>\n

    This example is particularly instructive for several reasons:<\/p>\n

      \n
    • the theorem itself is true,<\/li>\n
    • the proof outline looks standard,<\/li>\n
    • each sentence is individually reasonable,<\/li>\n
    • and the error occurs only at the level of algebraic<\/li>\n<\/ul>\n

      6. A STANDARD EXAMPLE MISCLASSIFIED<\/p>\n

      Prompt.<\/strong><\/h3>\n

      Give an example of a finite modular lattice that is not distributive. Name a standard lattice, describe its diagram, and explain why it is modular but not distributive.<\/p>\n

      ChatGPT response.<\/strong><\/h3>\n

      A classical example is the lattice N5<\/sub> (the pentagon lattice). It has elements 0, a, b, c, <\/em>1 arranged so that 0 is below a <\/em>and b<\/em>, a <\/em>is below c<\/em>, and both b <\/em>and c <\/em>are below 1. This lattice is modular but not distributive.<\/p>\n

      Modularity follows from the fact that there is no sublattice isomorphic to M3<\/sub>. Non-distributivity is witnessed by the failure of the distributive law for the triple (a, b, c<\/em>). Thus N5<\/sub> is a finite modular, non-distributive lattice.<\/p>\n

      Analysis. <\/strong>The classification is wrong. Among the two classical five-element lattices, only M3<\/sub> is modular and non-distributive. The pentagon lattice N5<\/sub> is the smallest nonmodular <\/em>lattice.<\/p>\n

      The argument compounds this error. It \u201cproves\u201d modularity of N5<\/sub> by invoking the absence of an M3<\/sub> sublattice, even though N5<\/sub> is the standard counterexample to modularity. The appeal to \u201cno M3<\/sub>\u201d sounds authoritative but is factually false.<\/p>\n

      This example illustrates a different kind of failure. The response has the following features:<\/p>\n

        \n
      • uses familiar notation (M3<\/sub>, N5<\/sub>),<\/li>\n
      • cites a standard characterization (forbidden sublattices),<\/li>\n
      • and gives a fluent informal<\/li>\n<\/ul>\n

        Yet it mislabels one of the best-known small lattices in the subject.<\/p>\n

        7. CONCLUSION<\/p>\n

        In every example above, the ChatGPT response is fluent, confident, and plausible. In every case, it is incorrect.<\/p>\n

        Verification is therefore not optional. It is an essential part of any serious mathematical use of ChatGPT.<\/p>\n

        [1] An element p <\/em>\u2208 L <\/em>is called join-prime <\/em>if, for all x, y <\/em>\u2208 L<\/em>,<\/p>\n

        p <\/em>\u2264 x <\/em>\u2228 y<\/em>\u00a0\u00a0\u00a0\u00a0 <\/em>implies that\u00a0 p <\/em>\u2264 x <\/em>or p <\/em>\u2264 y.<\/em><\/p>\n","protected":false},"author":11,"template":"","section":[338],"keyword":[],"class_list":["post-21086","article","type-article","status-publish","hentry","section-contributed-articles"],"toolset-meta":{"author-4-info":{"author-4-surname":{"type":"textfield","raw":""},"author-4-given-names":{"type":"textfield","raw":""},"author-4-honorific":{"type":"textfield","raw":""},"author-4-institution":{"type":"textfield","raw":""},"author-4-email":{"type":"email","raw":""},"author-4-cms-role":{"type":"textfield","raw":""}},"author-3-info":{"author-3-surname":{"type":"textfield","raw":""},"author-3-given-names":{"type":"textfield","raw":""},"author-3-honorific":{"type":"textfield","raw":""},"author-3-institution":{"type":"textfield","raw":""},"author-3-email":{"type":"email","raw":""},"author-3-cms-role":{"type":"textfield","raw":""}},"author-2-info":{"author-2-surname":{"type":"textfield","raw":""},"author-2-given-names":{"type":"textfield","raw":""},"author-2-honorific":{"type":"textfield","raw":""},"author-2-institution":{"type":"textfield","raw":""},"author-2-email":{"type":"email","raw":""},"author-2-cms-role":{"type":"textfield","raw":""}},"author-info":{"author-surname":{"type":"textfield","raw":"Gr\u00e4tzer"},"author-given-names":{"type":"textfield","raw":"George"},"author-honorific":{"type":"textfield","raw":""},"author-email":{"type":"email","raw":""},"author-institution":{"type":"textfield","raw":""},"author-cms-role":{"type":"textfield","raw":""}},"unknown":{"downloadable-pdf":{"type":"file","raw":"","attachment_id":null},"article-toc-weight":{"type":"numeric","raw":"99"},"author-surname":{"type":"textfield","raw":"Gr\u00e4tzer"},"author-given-names":{"type":"textfield","raw":"George"}}},"_links":{"self":[{"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/article\/21086","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/article"}],"about":[{"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/types\/article"}],"author":[{"embeddable":true,"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/users\/11"}],"version-history":[{"count":21,"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/article\/21086\/revisions"}],"predecessor-version":[{"id":21114,"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/article\/21086\/revisions\/21114"}],"wp:attachment":[{"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/media?parent=21086"}],"wp:term":[{"taxonomy":"section","embeddable":true,"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/section?post=21086"},{"taxonomy":"keyword","embeddable":true,"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/keyword?post=21086"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}