Reality and Gödel’s Incompleteness Theorems

0. Intro

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.
― Albert Einstein

Remember the promise you were made in the previous essay? To refresh your memory, I had boldly promised—ahem, clearing my throat now—that not only would you get concise, hard-hitting essays going forward, plus the very next one (the one you’re reading) will be from a guest writer, a terrific wordsmith and, far more importantly, a deep, clear-eyed thinker.

So here we are. Your wait is over.

The guest essay which follows—complete with British spellings—is by Dr. Saqib Riaz Qazi, a distinguished dental surgeon. My mind and heart are filled with joy at the mere mention of Saqib, a childhood friend. I’ll keep it brief, given my essayist propensity to, um, digress. But I feel compelled to mention that, during high school, he was my mentor in Physics. Thanks to Saqib, I had numerous light bulb moments as the essence of science began sinking into the cranium of yours truly: His knack for conveying the splendor of Physics opened up the world of science—and engineering—for me. Oh, and I remain impressed to this day by his wizardry in the area of hobbyist electronics. And we exchange notes on quantum mechanics and stuff like that.

With that intro, allow me to recede into the background, and give to you our guest essayist’s take on reality and Gödel’s Incompleteness Theorems. Take it away, Saqib.

1. The Basis

Either mathematics is too big for the human mind, or the human mind is more than a machine.
— Kurt Gödel

Mathematics, the basis for all science, is “incomplete”, as demonstrated by Gödel’s incompleteness theorems. Essentially, this means that, in mathematics—and other logical systems—there are always going to be unprovable truths. Put another way, if something or a system can speak about itself or describe itself or analyse itself, it will find unprovable truths. This is consistent with a definition of faith—you know something to be true, but cannot prove it using conventional “logic”.

2. The Endeavour

Pleasure soon exhausts us and itself also; but endeavor never does.
— Richter

An endeavour to assign a meaning to life through science—and hence mathematics, since all “scientific” explanations of reality boil down to abstract mathematical constructs—bows to the absurd and to the indescribable.

3. Coda

There can be no excess to love, none to knowledge, none to beauty, when these attributes are considered in the purest sense.
– Emerson

The beauty of mathematics, like all existence, may be described as a reflection of the one indescribable reality:

The seven heavens and the earth and those in them declare His glory. And there is not a single thing but glorifies Him with His praise, but you do not understand their glorification… (The Quran 17:44 )

4. Citations

The real voyage of discovery consists not in seeking new landscapes, but in having new eyes.
– Marcel Proust (in Remembrance of Things Past)

  1. As quoted in Topoi : The Categorial Analysis of Logic (1979) by Robert Goldblatt, p. 13
  2. Stanford Encyclopedia of Philosophy
  3. On the Philosophical Relevance of Godel’s Incompleteness Theorems Panu Raattkainen Dans Revue Internationale De Philosophie 2005/4
    (n° 234), PAGES 513-534
  4. How Gödel’s Proof Works
  5. Merriam-Webster Dictionary
  6. Stephen Hawking, A Brief History of Time: ”Even if there is only one possible unified theory, it is just a set of rules and equations. What is it that breathes fire into the equations and makes a universe for them to describe? The usual approach of science of constructing a mathematical model cannot answer the questions of why there should be a universe for the model to describe. Why does the universe go to all the bother of existing?

5. So You Want (Even) More?

When I have a little money, I buy books; and if I have any left, I buy food and clothes.
― Desiderius Erasmus Roterodamus

With the citations done—having cited that Erasmus quote, a personal fav—we are done for now. I sure enjoyed re-reading Saqib’s sterling essay, and hope that you, too, did.

We’re well into our brevity mode now, having taken a cue from the “less is more” movement, which you would’ve gathered from the previous essay.

Oh, should you wish to read up on Gödel’s Incompleteness Theorems, I can recommend a fine book devoted to that stuff: Incompleteness: The Proof and Paradox of Kurt Godel.

Finally, and paraphrasing from memory an intriguing sayingWhere angels fear to tread, fools rush init’s been said, analogously enough, that “Nature abhors a vacuum, and rushes in to fill the void.” Yep, much as empty spaces get snapped up when you’ve got prime real estate at hand.

Till next time, you could perhaps take a peek into rounding out this incompleteness business by, say, filling some empty spaces. Hmm… We’ll see about that, fools, though, we certainly are not, though realtors some of us may be.

One comment

  1. Superb post! Even though mathematical thinking is mysterious to me, I really enjoyed reading the post and Saqib bhai’s contribution is very thought provoking. Keep up the mind bogglingness. BTW this is Zainib, not Akram commenting!

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