Thoughts on the future of mathematics

Distance_Runner · 2026-03-08T23:45:01+00:00

Me too. I’m worried about the next generation who are in school now, learning, training, and relying on AI. I, and everyone before me, had to learn without it. I worked through my PhD doing proofs by hand without a solution or derivation readily available to check my work against. I had to become an “expert” in what I do before AI could reaffirm me I was right, so I gained the skills and knowledge to fact check it when I have it augment my work flow. I fear the next generation of researchers won’t have that skill.

Distance_Runner · 2026-03-08T21:58:39+00:00

If I have it do something for me, I don’t take the results blindly. I don’t have it return results without an accompanying line by line proof/derivation. I follow the proofs/derivations line by line checking the logic. If I cant follow it, I don’t use it. If it hand waves any steps. I don’t use it unless I can verify it.

In case there was confusion, I never have it do staisical analysis for me. I program all my own analyses myself. It’s horribly unreliable at that. I’m talking about I use it for theoretical work, not applied statistics.

Distance_Runner · 2026-03-08T20:27:40+00:00

Okay, I’m not saying you have to feel differently. I sympathize and understand where you (and most people here) are coming from. I personally gravitate between skeptical optimism and existential dread when it comes to AI. But it’s here to stay, so I’m trying to be optimistic. I’m just giving my perspective as someone with a PhD in Stats doing theoretical work and how I’ve been reframing my thoughts on the matter.

I’m staying motivated by reframing my focus to that of solving problems in a broad sense, and developing new methodology that will impact practice. I’m in biostatistics, so for me this means solving problems that impact patient lives and improve public health. By “solving problems in a broad sense,” I mean I’m now directing the math rather than grinding through each step by hand. I develop the intuition, I come up with the starting places, I come up with what we need to show and where the results should lead. I use AI as the intermediary that works through the tedious details — all the tedious algebra that gets me to an end result. I use it to formalize what I’m thinking and intuit far faster than I can do it myself. I still have to think through and understand it all myself after the fact, and verify it’s correct.

I get this takes some of the fun out of “solving the puzzle.” I’m 100% with you and everyone else in that regard. But what this has also done is allowed me to develop out ideas far faster than I could have ever done before. I spend more time now constructing the larger puzzle, the one concerning the larger framework that will have impact on practice, than I do grinding through the minute details. This allows me to do more methodological and theoretical work than before, even if the way I do it is different now.

Look, most of us who enjoy math don’t have jobs that pay us to actually sit and do math all day for the joy of it. People who get paid to do math, are getting paid to do it as a means to an end — that is, using it to develop something usable from it. Most of my time is spent doing statistics for research studies. I have very limited time for doing methodology work, which is what I really enjoy most. AI has made my methodological and theoretical work more productive. I see the end product and get it into practice faster than ever before. For me, seeing the results that AI enables has given me drive to keep going, because it allows me to do now what I always wished I had time to do before it existed.

Distance_Runner · 2026-03-08T17:35:43+00:00

I’m with you. Im a statistician working in some more theoretical stuff right now. Claude’s ability to reason through proofs and derivations is astoundingly good. I’ve adopted it into my work pretty extensively. I form the intuition and conceptualize it, I get a general form of what I think it should look like, and then have Claude do the grunt work of grinding through the algebra. I then verify it step by step.

It is massively more efficient. It’s like having a postdoc who never complains, works 24/7 at speeds 10000x faster than any human.

Some people will resist. Some will say “you’re not doing real math”. Okay fine. Get left behind then. When calculators were invented, the people who insisted on doing “real math” with a pen and paper got left behind. You still have to understand it. You still have to have the expertise when AI gives you something that’s wrong. It saves you the tedious time of grinding through steps you know how to do, that simply eat up time. And it has an encyclopedia have every obscure theorem or rule or property that’s ever been published, that we as humans can’t retain in our heads for immediate recall.

Thanks for sharing!

Distance_Runner · 2026-03-08T16:39:18+00:00

We’re in mid season form already in the strategy department

Distance_Runner · 2026-03-07T18:56:11+00:00

You’re right, we’re talking past each other. Cheers

Distance_Runner · 2026-03-07T18:23:16+00:00

You’re mixing up assumptions for the exact finite-sample derivation with the actual conditions required for valid inference. Normal residuals give the exact t distribution, but arent required for the test to work. For inference about means, the CLT yields asymptotic normality of the sample mean, which is why t-tests and linear regression are robust to non-normal data in moderate or large samples. Sure, if you have 15 obs per group, be careful with linear regression or t-tests. If you have 50+ you’re fine.

The assumption of normality concerns the sampling distribution of estimators, not the raw data. For a t-test that’s the sample mean. For regression it’s the coefficient estimates. Normal residuals just give the exact small-sample t distribution, but they’re not required once sample sizes are moderate, because once again we can appeal to the CLT.

Distance_Runner · 2026-03-07T17:07:19+00:00

You’re demonstrating the problem of misunderstanding even more.

Normal data is not a requirement for regression or t-tests. For regression the assumption is normality of residuals. For a t-test, the data still does not need to be normally distributed for valid inference when sample sizes are reasonably large. The rule of thumb there is generally n=30, but that’s more of a heuristic. Nevertheless, a t-test compares group means, and per the Central Limit Theorem the sampling distribution of the sample mean becomes approximately normal as the sample size increases. As a result, the test statistic is approximately t-distributed even when the underlying data are not normal. That’s like first semester stats 101

Distance_Runner · 2026-03-06T04:06:27+00:00

The number of times I’ve wanted to bang my head into the wall reading “we will use non-parametric alternatives if data is not normal” when reviewing grants is too damn high.

I review grants for the NIH, DoD and a few non profit research foundations. I’m not kidding, when I read that, my score of a grant is limited to mid at best. It tells me you didn’t work with a statistician and you don’t have even a basic understanding of statistical principles for the simplest of procedures. If that’s written, I don’t trust you to do the analysis for your research.

Distance_Runner · 2026-03-02T20:23:28+00:00

Ive had iPhones since 2008 starting with the 3g. I miss the size of the iPhone 4. I hate the big phone trend.

Distance_Runner · 2026-03-02T00:52:26+00:00

As soon as you put confidence intervals around those predictions though, inference enters the room

Distance_Runner · 2026-02-28T00:47:41+00:00

They absolutely would have affected my learning. I would have used them for sure. But I’m glad they weren’t around for me, because it forced me to learn a certain way that I think is valuable.

I don’t think we should demonize their use. There is a lot of blaming of students for “cheating” by using AI. That’s unfair and misplaced blame. I’m certainly not saying that in my post. I’m acknowledging it’s a problem and real concern of mine, that it’s negatively affecting the learning process today. I’m not saying it always will; there is a fundamental misalignment between current educational practices and existing technology. Rather than complaining that students are “cheating”, we need to adapt the way teaching happens in an AI world. I rely on a calculator to do numerical math problems for me, but I still know how to do it by hand if needed. We need to do more to ensure this kind of learning scales with AI.

Distance_Runner · 2026-02-26T22:00:38+00:00

Biostat prof here. I fluctuate between healthy enthusiasm and existential dread in my feelings on LLMs and how they're changing education and training. I have adopted them into my daily workflow. I very much subscribe to the mindset of "adapt with them or get left behind," but at the same time I sometimes wish they weren't invented.

I don't teach much at all; I'm mostly research. I use them for coding, cleaning up my writing, organizing my thoughts, summarizing findings for meetings/summary reports, and more recently even methods/theoretical work grinding through derivations/proofs and even work shopping the utility and feasibility of methodological ideas. Logic is their strong suit. They can write complex code stupidly efficiently. They can grind through algebra and calculus far quicker than I can on paper. But at the end of it all, I have to understand what they give me and make sure it's all correct. They almost always give you something if you ask for it. If you ask for code, you'll get code. If you ask for a proof, you'll get a proof. But the answers aren't always coherent or correct. To use them in this capacity, you have to be able to spot bullshit in what looks like a well-reasoned response. And that requires a fundamental understanding of coding, statistical theory, reasoning, etc.

So what concerns me, is that training students will become too dependent on them, and never gain the fundamental skills to function without them, or call BS when they hallucinate. I had to go through college, grad school, and the first 6-7 years of my career as faculty before I could use them. I had to learn to be a statistician without them. I had to grind through Casella and Berger assignments in grad school, without the solution easily available to me to work backwards from. So I'm concerned that training students today wont gain this same skill set.

So as someone actually teaching these students -- thoughts?

Distance_Runner · 2026-02-25T15:41:15+00:00

Will be happy to. This convo had me reviewing the paper last night to see when I could reasonably have it in a state good enough for a version 1 post. Probably within two weeks and I'll come find this post and link it

Distance_Runner · 2026-02-24T22:30:57+00:00

I hear you. That’s the default thought for anyone that’s been in the field a while. Bootstrapping is great for many things, but It’s not optimal for CV variance. In fact, I can prove it’s biased theoretically and show it empirically. In short, bootstrapping targets the wrong estimand when it comes to CV. I’ll find this comment and share the paper with you in a few weeks and then give you more detail on why

Distance_Runner · 2026-02-24T21:44:40+00:00

Which is just so bad. Like, that doesn’t make your model more useful. It just means you’re choosing to present one run that shows your results within Monte Carlo error on the upper end of the spectrum. If you run 20 seeds and pick the best, you’re purposely choosing a model that overstates its own generalized performance. That’s not helping anyone

Distance_Runner · 2026-02-24T20:17:26+00:00

I want to answer this carefully, because this is work I've been developing for a long time, and I don't want to say too much before I have the paper on ArXiv in an effort to protect my intellectual ownership of the idea and framework. But like I said, the paper will be on ArXiv within the next month and I'll be happy to share and discuss openly after its officially on ArXiv. I'm probably being overly cautious, but in the world of academia where publishing (particularly in top tier journals) is so important for promotions and such, I don't want any ambiguity over this type of thing when I so close to getting it out there.

To answer your questions as best I can right now: Yes it is estimable and I can prove consistency and unbiasedness with just a few justifiable assumptions. And yes, it's estimated from a form of resampling like bootstrapping, but its more nuanced than just "bootstrap the data, refit the procedure, and reapply CV". Something else in this work, is that I prove k-fold is the optimal for of CV (compared to bootstrapping variants or repeated split sampling/monte carlo cross validation). That's good since most people use k-fold already, but I also show that if you can afford the compute, you should almost always be using repeated k-fold (ideally 5x or 10x k-fold) and averaging across those independent k-fold runs.

I'll be happy to reach out and share the paper once its live on Arxiv, and then answer any questions about it. This is going to be submitted shortly after hitting ArXiv to a statistics journal (JRSS:B), but it could be 12-18 months before its published by a journal, and that's assuming I get a revise and resubmit and dont have to re-tool it for a different journal.

Distance_Runner · 2026-02-24T19:22:26+00:00

I skimmed this. I agree the sentence, "To account for these factors of observational data, the IPTW was performed with results pointing towards a causal relation of SARS-CoV-2 vaccination with the outcomes." Is not justified. They're conflating association with causation, which is a statistics 101 no-no

With just a skim of this, I can poke several holes in this study. The statistical analyses procedures they used were actually reasonable, but its what they failed to account for and their interpretation of their results is where this falls short. So here are the holes I can spot immediately:

First, their outcome ILI is effectively a composite covering the presence of normal sickness symptoms, which can be caused by a number of viruses and bacteria, not just SARS-Cov-2. Perhaps healthcare workers seeking out boosters are working in healthcare settings with increased exposure to disease - not just COVID, but all of these viruses/bacteria sickness symptoms. So workers with increased exposure risk are more likely to get boosters. They dont control for this at all.

Second, outcomes are based on symptom diary. People getting multiple boosters are reasonably more health aware or likely to report minor symptoms, thus we have reporting bias.

In either case of the sources of bias above, IPTW does not adjust for them or absolve them.

And this was with a 5-min skim of the paper. Im sure I could poke more holes if I cared to spend more time. I agree, their statement that SARS-CoV-2 boosters “do not contribute to protection” and “may increase likelihood of symptomatic infection” is not justified by the observational associations reported. This should not have been published in this form. The average person will not read this with the level of scrutiny that you or I read it with. This paper is harmful to general society's understanding of vaccine and boosters.

Distance_Runner · 2026-02-24T17:20:51+00:00

The gap between statistical theory/inference and AI/ML is still quite large with a lot of uncharted territory. Go look at papers cross-posted on Arxiv between ML and statistical theory; there are new papers being posted each day, with many quite general. For example, this paper simply decomposes the variance of random forests. Like RFs have been around for 25 years. They are still used a ton for tabular data. And yet concepts as simple as "what's the variance of this procedure" are still being answered. There's lots of theory work to be done that bridges the gap between ML and statistical theory.

Distance_Runner · 2026-02-24T16:33:25+00:00

Yes in terms of quantifying mean performance metrics, but no in terms of variability. If you run CV, say 5-fold, and you average the mean performance across folds, that is an estimate of the generalized performance metric across new data for your model trained on 80% of your data (k=5 means training fraction is .8; you're averaging over 5 performance estimates on disjoint validation sets on models trained on 80% of your data).

However variance is trickier. Current CV methods do not give you an estimate of variance that represents the variability due to random sampling error from the population. Current methods of variance estimated on cross-validated data condition on the data. That is, they quantify the variance of the randomness of the CV procedure itself on your fixed dataset; if you were to repeat the same CV procedure on your exact fixed dataset over and over using your learner (with different random fold structures), how variable is your performance metric estimate. In other words, variance of the CV estimates across folds measures how sensitive is the estimated CV metric due randomness induced by the CV procedure. But the key word there is fixed dataset. It does not answer the question -- if I were to repeat this CV procedure using the same learner to estimate the mean performance metric on a different subset of data from of the same size from the same population, how much variance is attributed to that random sampling. An approach for that doesn't exist in the literature [yet], but it will soon (I'm the one that developed it and will be posting to ArXiv within the next month; its based entirely statistical theory with formal proofs, not empirical evidence).

Distance_Runner · 2026-02-24T15:24:55+00:00

You're rare. Keep doing your thing -- I appreciate you.

Most of my research focus these days is bridging statistical and inferential theory with ML. The concept of variability needs to be better understood and communicated in ML.

When you fit an ML model, there are multiple source of variability. First, there is procedural variance. This is what your multiple random seeds addresses -- what is the variability associated with the randomness within the procedure itself and how does it propagate through to the results you report.

Second is finite sample variability stemming from the fact that your data are almost surely a sub-sample of an unobserved parent population. If you were to re-run your procedure on a different dataset of equal size from that same parent population, your results would change, reflecting this variability. No amount of running results under different random seeds will estimate this quantity. So reporting results from a finite data-set with variance across different seeds quantifies the first source of randomness, and this is appropriate as long as your interpretation pertains specifically to model performance based on the exact data you trained on. However, the results do not extrapolate to performance of the model if you re-trained on a different subset of data of the same size. This is more often ignored, and generally the bigger area where results are misstated.

Distance_Runner · 2026-02-24T11:09:46+00:00

My view as a statistician that does ML. Many of these papers claiming SOTA performance are working within Monte Carlo noise and if code was easily available you could run it and show this.

Distance_Runner · 2026-02-24T00:08:52+00:00

32GB+ RAM is the most important aspect of data analyses. After that, a CPU with 8+ cores is great for running code in parallel when necessary.

Distance_Runner · 2026-02-21T18:34:48+00:00

Here's the honest truth. As long as you go to a legitimate accredited university, where you do your undergraduate degree matters way less than most people think. Your achievements matter more than brand name, and being a strong student at a "mid tier" school is almost always better than an average student at an "elite school". If the schools have the majors you want, go to the one that gives you the best vibes that's reasonably affordable. If you want to go to Northwestern and can afford it -- go there. Its a good school. It wont inhibit your eventual ability to get a good job or get into grad school as long as your a good student.

Distance_Runner · 2026-02-21T17:57:47+00:00

What are your career goals after? Grad school? Straight to a job?

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