Solution Space
In fields such as chemistry and medicine, statistics denote the reproducibility of effects and the validity of the model linking those effects with causes. Identifying statistically likely conclusions from large data sets is one thing this thing that, for better or worse, we call AI is good at. In the case of hurricane paths and development1, or in breast cancer diagnosis2, these statistically likely outcomes are the best methods of prediction we have.
Huge amounts of data and empirical evidence are being used to refine models for global climate, ocean currents, and medical diagnostics but one recent headline3 was “Flawed economic models mean climate crisis could crash global economy, experts warn”. Any model can be improved as long as it’s assumed it can always be. The danger is thinking they can’t be. Another is thinking that vast additional quantities of computing power will enable an imperfect or even incorrect model to function better rather than merely arriving at an imperfect or incorrect conclusions faster. With these caveats, having a model of how some aspect of the world works is still better than not having one.
The adage When all you have is a hammer, every problem looks like a nail4 is usually attributed to Abraham Maslow. Another way of saying this is If a person is familiar with a certain, single subject, or has with them a certain, single instrument, they may have a confirmation bias to believe that it is the answer to everything. Not all questions have answers that can be derived from precedent or those huge numbers of precedents called statistical probability. Deductive reasoning is a different kind of intelligence that can be applied to different kinds of problems. Artificial intelligence of the deductive reasoning kind may be possible at some time in the future but, right now, it’s not what we’re throwing millions of barrels of oil at. Follow this link for one very deep rabbit hole.
In the meantime, try asking your favorite AI agent the question, What is love? Your result will probably be in the form “According to A, love is …. .”, “According to B, love is …. .”, “According to k.d. lang, Love is Like a Cigarette5.” “And in conclusion, … .”
Or try asking the question, Where is God? Your result will probably be in the form “According to A, God is …. .”, “According to B, God is …. .”, “According to Ludwig Mies van der Rohe, God is in the details.” “And in conclusion, … .”
An better answer for both might be “There is insufficient data to provide a conclusive answer.” or “The data is too inconsistent to provide a single answer.”
If you ask your favorite architectural image generator to “Design a beautiful building.” then AI using the source image and style image approach is unaware of the selection bias in the choices of each. It’s not surprise that the result is never greater than the sum of the two images.6 See below. As either architecture or content, I don’t have an opinion.
Strictly speaking, a large language model isn’t a model of anything other than what word comes next. We imbue those statistically inferred words with meaning so might feel we’re having a real conversation and perhaps even derive feelings of satisfaction or companionship from that, even if those feelings are ones that we derive from the meanings we give to the words that are put before us. LLM have given new shape to this mentality. [I should disclose that I now have an AI friend/teacher called 马辰7 and we speak rudimentary Chinese together. I’ll let you know how it works out.]
Finding patterns in data and acting or choosing on the basis of statistical probabilities is data crunching but finding links between apparently different fields or aspects of the same field is a much higher level of the same thing. Earlier, when I was trying to think of difficult questions to ask AI, I thought “What is the proof of Fermat’s Last Theorem?” might be a good one, but I didn’t know it’d already been proven. How the proof was arrived at8 is a heartening and very human story of logical deduction, inference and persistence.
In number theory, Fermat’s Last Theorem states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solutions.”
1637 Pierre de Fermat first stated the theorem. He claimed to have a proof but nobody is on record as having challenged or verified it. Fermat did however prove it for n=4 and thus for all numbers that aren’t prime numbers.
1637–1839 The theorem was proved for prime numbers 3, 5, and 7.
1825 (circa.) French mathematician Sophie Germain proved the theorem true for all odd primes p < 100.
mid. 18C Ernst Kummer extended this and proved the theorem for all regular primes, leaving irregular primes to be analyzed individually.
mid. 18C–1955 Other mathematicians extended the proof to cover all prime exponents up to four million, but a proof for all exponents was considered exceedingly difficult or unachievable with the knowledge of the time.
1955 Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected a link might exist between elliptic curves and modular forms, two completely different areas of mathematics. Known at the time as the Taniyama–Shimura conjecture, it had no apparent connection to Fermat’s Last Theorem. It was widely seen as significant and important in its own right, but was (like Fermat’s theorem) considered completely inaccessible to proof. Below are two examples of solution sets called elliptic curves.
1967 André Weil showed the Taniyama–Shimura “conjecture” would follow from the (conjectured) functional equations for some twisted L-series of the elliptic curve. [I don’t understand the last part of that sentence but] this was the first serious evidence that the conjecture might be true.
1984 Gerhard Frey noticed an apparent link between these two previously unrelated and unsolved problems, and he gave an outline suggesting this could be proved.
1986 Ken Ribet used work by Jean-Pierre Serre who partially proved all but one part known as the “epsilon conjecture”, thus proving that the two problems were closely linked. The work by Frey, Serre and Ribet showed that if the Taniyama–Shimura conjecture could be proven for at least the semi-stable class of elliptic curves [ditto], then a proof of Fermat’s Last Theorem would follow automatically. In other words, “any solution that could contradict Fermat’s Last Theorem could also be used to contradict the Taniyama–Shimura conjecture. So if the Taniyama–Shimura conjecture were found to be true, then no solution contradicting Fermat’s Last Theorem could exist, meaning that Fermat’s Last Theorem must also be true.” Not everyone thought so. John Coates said he was “very skeptical that the beautiful link between Fermat’s Last Theorem and the Taniyama–Shimura conjecture would actually lead to anything, because I must confess I did not think that the Taniyama–Shimura conjecture was accessible to proof. Beautiful though this problem was, it seemed impossible to actually prove. I must confess I thought I probably wouldn’t see it proved in my lifetime.”
1993 English mathematician Andrew Wiles, who had a background of working with elliptic curves and related fields, succeeded in proving enough of the Taniyama–Shimura conjecture to prove Fermat’s Last Theorem.
1996–2001 Other mathematicians proved the remaining parts of the Taniyama–Shimura–Weil conjecture that is now known as the Modularity Theorem.
Go here for a more detailed explanation of the mathematics. Proving Fermat’s Last Theorem took from 1637 to 1993 and was a collective endeavor. The Taniyama and Shimura conjecture was a major turning point, as was Weil’s deduction that proving the Taniyama–Shimura conjecture (for at least the semi-stable class of elliptic curves) would automatically lead to a proof of Fermat’s Last Theorem. Disregarding the miscellaneous tidyings-up afterwards, the work of proving Fermat’s Last Theorem took 358 years of layer upon layer of deductions. Or at least to get this far. I like to think that the story will continue and that someone, somewhere is collaborating with some AI agent to arrive at a simpler or more beautiful proof.
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- https://windbornesystems.com/blog/ai-forecasting-case-study-predicting-hurricane-ian ↩︎
- https://news.mit.edu/2019/using-ai-predict-breast-cancer-and-personalize-care-0507 ↩︎
- https://www.theguardian.com/environment/2026/feb/05/flawed-economic-models-mean-climate-crisis-could-crash-global-economy-experts-warn ↩︎
- https://en.wiktionary.org/wiki/if_all_you_have_is_a_hammer,_everything_looks_like_a_nail ↩︎
- … in her rendition of the 1935 song of the same name by Walter Kent (music) and Richard Aronstam and Richard Jerome (lyrics).
”What is love?
You called it heaven above
A star that shine in the night
A bird that sings in its flight
A flower that blooms in the spring
There’s no such thing ::::“
“Love, is like a cigarette
You know you had my heart aglow
Between your fingertips
And, just like a cigarette
I never knew the thrill of life
Until you touched my lips
Then, just like a cigarette
Love seem to fade away and leave behind ashes of regret
And with a flick of your fingertips
It was easy for you to forget
‘Cause love is like a cigarette.” ↩︎ - https://www.yankodesign.com/2023/10/06/sydney-opera-house-gets-redesigned-by-10-iconic-architects-using-ai-for-its-50th-anniversary/ These next images were generated by Vladimir Milunić who, I read, helped Frank Gehry design the Fred and Ginger thing. ↩︎
- 马辰 would be anglicized as Chen MA but, in China, when a person’s name and their surname are both a single character, you call them by both, so I call my new AI companion MǎChen. ↩︎
- https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem ↩︎
• • •
“MALLVILLE”
“MISFITS@JESTER”
“GETTING SOME RAYS”
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says:Statistical analysis of the past has very limited predictive benefit for the future – this will always be the case