Lost in Math?

<i>&gt; About 10 years ago, in the wake of the 2008 financial crisis, the Nobel Laureate economist Paul Krugman made the same point with respect to economics and mathematics in an influential article titled &quot;How Did Economists Get It So Wrong?&quot; His main answer was: mistaking mathematical beauty for truth. &quot;As I see it,&quot; wrote Krugman, &quot;the economics profession went astray because economists, as a group, mistook beauty, clad in impressive-looking mathematics, for truth.&quot;</i><p>I appreciate the point the article is trying to make, but I think this example is shoehorned in. You can misuse math without it being because you&#x27;re &quot;seduced by the beauty&quot; of it.<p>I do agree with the author&#x27;s example in physics. I have seen a lot of beautiful math in physics; look at lie algebras, monstrous moonshine and representation theory. Quite a bit of modern physics PhD dissertations are actually just math dissertations, and the same holds for a significant amount of new research in the field.<p>On the other hand I haven&#x27;t seen that in finance. Highly exotic (read: &quot;beautiful&quot;) mathematics is extremely rarely used in financial engineering. Pricing derivatives is decidedly mundane work compared to the brain-meltingly abstract mathematics deployed in high energy particle physics research. That&#x27;s not to say it isn&#x27;t difficult - it is difficult! But difficulty is better described by the word &quot;complex&quot; rather than &quot;beautiful&quot;, and then <i>of course</i> financial engineering is complex. Then we should be talking about how getting mired in complexity can be bad for accountability and transparency.<p>This is a different thesis than the one presented by the author. Being led astray because you&#x27;ve built extremely brittle financial products using layers of complicated math is <i>not</i> the same as being more preoccupied with the elegance of a grand unifying theory than its agreement with reality.<p>But hey, maybe I&#x27;m just being pedantic. You can misuse mathematics in a lot of ways.

Related: Dijkstra&#x27;s comments on mathematics and CS - <a href="http:&#x2F;&#x2F;www.cs.utexas.edu&#x2F;users&#x2F;EWD&#x2F;transcriptions&#x2F;EWD12xx&#x2F;EWD1243a.html" rel="nofollow">http:&#x2F;&#x2F;www.cs.utexas.edu&#x2F;users&#x2F;EWD&#x2F;transcriptions&#x2F;EWD12xx&#x2F;EW...</a>.<p>I&#x27;ve personally been getting a lot of satisfaction from learning Haskell and seeing how the functional programming community is taking ideas from abstract mathematics, like category theory, and is turning them into practical ways of thinking about programming.

<i>But complexity theory aims at describing the performance of A over the space of all problem instances and it does so by abstracting away from individual problem instances.</i><p>I appreciate the effort to extend the story into CS, but I wonder if you have to be familiar with the particular work he&#x27;s alluding to. The charge (as leveled against theoretical physics) is not that some people do pure mathematical work for the sake of beauty. The charge is that people who are <i>supposed</i> to be applying mathematics to reality are instead prioritizing mathematics and neglecting reality. To extend the analogy to CS, he must be talking about researchers supposedly trying to model real systems but instead just chasing beautiful math, but he isn&#x27;t specific. Is it obvious to people in the know who or what he&#x27;s talking about?<p>For practical programmers, I think the problem is the reverse of being &quot;lost in math.&quot; Practical programmers use extremely general theoretical results because they <i>don&#x27;t want to do</i> math, not because the math is more beautiful. If they applied the information they know about their particular problem, they could get more useful mathematical results, but since they want to stay as far away from (doing) theory as possible, they use whatever facts they remember from class, which are ironically the most purely theoretical ideas because those are the simplest and easiest to remember.

I haven&#x27;t been in the developer industry for too long, but excepting the haskell community, I would say that the way CS tends to treat math is as guardrails, as in, &quot;you can&#x27;t do that because of the halting theorem&quot;. &quot;you might be butting up against computational complexity if you try doing it this way&quot;. &quot;reconstruction of this data shard is impossible because you don&#x27;t have enough points to determine the equation&quot;.<p>In the FP communities, I <i>do</i> sometimes see people overoptimize for TCO. Your datastructure is never going to be more than 10-100 deep. Don&#x27;t worry about it. Just write the most legible recursive algorithm, not the most performant.

<i>But the seductive power of mathematical beauty has come under criticism lately. In Lost in Math, a book published earlier this year, the theoretical physicist Sabine Hossenfelder asserts that mathematical elegance led physics astray. Specifically, she argues that several branches of physics, including string theory and quantum gravity, have come to view mathematical beauty as a truth criterion, in the absence of experimental data to confirm or refute these theor</i><p>Her criticism has gotten more attention than justified by its merits. No one has argued that beauty holds precedent over truth

&gt; the theoretical physicist Sabine Hossenfelder asserts that mathematical elegance led physics astray<p>I feel the other way around: Applied mathematicians and physicists led the pure mathematicians astray. But I say this for a different reason. I feel that physics has become convoluted with a plethora of theories where this same beauty is interpreted wildly differently by different people. In other words, people all have different ways of thinking, different notions of beauty, and ultimately, this manifests into different (competing) notions in physics. These may even be equivalent notions and they may embody the desired perspective on beauty, but instead of consolidation there is extended differentiation.<p>My point is that physics has become ugly exactly because of physicists&#x27; ignorance towards mathematical elegance in favour of personal beauty. I don&#x27;t think physics can become consolidated without exactly a stark appreciation for elegance.<p>IMO this is why category theory took so long to start appearing in physics: The physicists are caught up in their own idea of <i>beauty</i> rather than the mathematical tradition of finding the minimal sufficient proofs and theories (which I call elegance).

Coming from a mathematics background, I personally do not care so much for the &#x27;beauty&#x27; of mathematics, and am moreso interested in clarity of properly abstracting and insight to the resulting formal theory. I feel that physicists care more about such intuitive ideas than anybody else.<p>Regardless, you only can become lost in math if you have bad premises. Mathematics is a relative subject, abstracting the arbitrary of reality to axioms. And if the axioms do not hold, the theory is bunk. It will always be the case that more granularity is required in real-life situations. Mathematics is precise and sound; it&#x27;s not gospel.

If you count all the work in AI&#x2F;ML then the criticism has been overwhelmingly in the other direction, i.e. to much &quot;just trying stuff to see what happens&quot; and not enough &quot;really understanding what is going on&quot;. Always seemed like a weak criticism to me honestly. You can advance theory, or you can advance through experimental insight. Neither is the right or wrong path, just whichever seems like the best way to make progress given the state of current knowledge.

I have long felt like &quot;beauty&quot; in mathematics is just oversimplification.<p>Ironic that intelligent mathematics types get caught up in what could be analogous to socially hurtful stereotypes.<p>I am going to follow the author.

It&#x27;s terrible that nobody got punished for it, I hope Bitcoin will solve all of our problems soon.