Monday, December 7, 2015

Cogito Ergo Sum (ii)


Let's try something different. Show that

            1       1         1
(1)       ----- - ----- < ---------
          1,000   1,001   1,000,000



Proof

             1       1      1,001 - 1,000        1
(2)        ----- - ----- = --------------- = ---------
           1,000   1,001   (1,000)*(1,001)   1,001,000

But 1,001,000 > 1,000,000, so

              1           1
(3)       --------- < ---------
          1,001,000   1,000,000

But equation (2) says that

  1       1         1
----- - ----- = ---------
1,000   1,001   1,001,000

Replacing in equation (3), we reach equation (1)

            1       1         1
          ----- - ----- < ---------
          1,000   1,001   1,000,000

Which is what we wanted to prove all along (often symbolised as QED: quod erat demonstrandum).

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The previous example (slightly abbreviated) was taken from "Proof in Mathematics, an Introduction", by James Franklin and Albert Daoud (1996, Quakers Hill Press, Sydney). It's their first example, in their first chapter and it's meant to provide an "easy and short" illustration of the concept of proof (particularly, mathematical proof, although logical proofs are not that different): it involves only basic arithmetic operations and it's developed, step-by-step, so that readers with little formal education can follow.

Attentive readers will notice that we start with equation (1) and end with equation (1). In a way, you can see this proof as the "legal", "valid", "lawful" steps needed to "build" equation (1). I invite readers to go through the proof, at their leisure. There is no rush.

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At this stage, after careful consideration, self-confident readers with an economics background (particularly, Austrian or Keynesian) may object disdainfully:
"But, but, but... you assume your conclusion!"
Well, duh.

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I suppose Keynesian readers, inspired by Joan Robinson, won't need this, but other readers, who may be interested on what proofs (particularly mathematical proofs) are, may find this article (and the links therein) useful:

"Why we want proof?", by Marianne Freiberger

Incidentally, you'll find that an insightful Anonymous -- maybe an online Keynesian? -- already commented. You've gotta love Anonymous(es) or is it Anonymice?

2 comments:

  1. I'm not sure where you're going here.

    You don't actually assume your conclusion: you don't take (1) as true. You start with (2) and use deduction to get to (1). You just state (1) as uncertain, as a goal. There's nothing even resembling circularity.

    Sadly, the Freiberger link appears to be broken.

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    Replies
    1. Hi Larry.

      "Sadly, the Freiberger link appears to be broken."

      It worked for me just a moment ago. Maybe the server sometimes gets slow?

      "You don't actually assume your conclusion: you don't take (1) as true. You start with (2) and use deduction to get to (1). You just state (1) as uncertain, as a goal. There's nothing even resembling circularity."

      The fact one is not assuming one's conclusion has never stopped Keynesian economists from accusing one of that. So, one might as well enjoy the ride. :-)

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      I was being ironic, my friend.

      Seriously now. I have been thinking about this:

      "(...) I do see a problem with your analysis.
      "You partly assume your conclusion to start: revenues are fixed at Y. If you cut wages and did not increase dividends, and kept your other assumptions the same, your revenue would fall short of Y and profits would not rise. Indeed, it is the increase in dividends—a profit source—that creates additional profits, not the wage cut. This can be seen if you rerun your example with a raise in dividends without any wage cuts. Thus, all you prove is that increasing dividends increases profits."


      (You might remember that from a previous post).

      I find it amazing how painfully unprepared to argue logically economists in general are and how oblivious they are of that.

      Particularly economists of the Keynesian variety. And I wonder if the examples Keynes and Robinson set are not at least partly behind that.

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