Ask HN: Is it possible to get PhD without completing bachelors

62 points by throwaway256256 ↗ HN
Hello! I have been an employed software engineer for a bit more than 8 years, after dropping out of a traditional bachelors degree program in 2012, due to financial and personal reasons.

I have been the “theory guy” in every job I’ve had in the last 6 years, and I would really like to be able to learn more theoretical math-and-computer-science-related subjects, since I am more financially stable and seeing a therapist regularly. I don’t have much interest in completing my gen-Ed’s, but I genuinely would like to learn more theoretical math/compsci, and I can only get so far buying used textbooks on eBay.

Is it possible to get a graduate degree without a bachelors (in the US)? If anyone here has done that, can you give any advice on how?

61 comments

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For all practical purposes I doubt it. I suppose there is the occasional genius who is able to obtain credentials without going the traditional route, but from what I've read these sort of situations are highly situational. Y ou can find a few existing threads on the subject with google.

I've heard of people getting into masters programs without and undergrad, often persuading a school too take into account their work experience in a respected field, but it's still not very common.

Also my understanding (I may he wrong) is that a PhD is not about the classes as much as the research you do.

> Also my understanding (I may he wrong) is that a PhD is not about the classes as much as the research you do.

That's correct. A PhD is a program that culminates in a candidate being able to defend a thesis based on his original research when challenged by a jury of reputable academics.

That's what a PhD boils down to: conduct research that is deemed acceptable and holds to scrutiny of fellow academics.

In fact, IIRC it's possible to complete a PhD program by having a series of papers accepted in reputable journals that follow a coherent theme, as long as they can be used to support a thesis.

In theory perhaps, but in practice no. To get a phd position at a reputable institution, you have to compete with people who have both Master's degrees and often have published multiple papers. Oh, and getting a phd position is the easy part - finishing it is the difficult one!
Can confirm. I'm stuck in dissertation procrastination mode. This summer, boredom has moved me forward on writing but you're 100% correct that landing a slot is much easier than fulfilling all of the requirements (the key piece being that pesky dissertation stuff!).
I would add that if anyone really wants and is capable of enduring a PhD program and capable of getting said PhD, completing a BA is by far the easiest and simplest part of the whole process. Thus there is really no reason to avoid getting those qualifications under your belt.

In the very least you might be forced to waste some part of your PhD program to learn 101-level stuff just to catch up with your starting point.

I have met one person who did it. Even then they were a certifiable genius, still had to jump through an enormous amount of bureaucracy with an enormous amount of sponsorship from many faculty whos grant funding was directly tied to this person working on their project.

So in practice, no.

If you want to learn more theory, I do think it's possible to apply and get an MS. Especially now since MS programs are bleeding people due to remote learning.

> Especially now since MS programs are bleeding people due to remote learning.

Can you elaborate on this? Are they just losing students? Why do you think it's happening?

Paying $40,000 to watch a professor give a dry Powerpoint talk over a box in Zoom ain't the deal universities think it is.

I can get a better education with a textbook and a Youtube for free, let alone real online courses.

Universities have neglected teaching-and-learning for the better part of a century, and with COVID19, it's coming back to bite most of them. For a lot of programs, enrollment numbers are dead. Professors have viewed teaching as what they have to do in order to do research, and many approach it with all the vigor and intellectual rigor that entails. You show up. You talk at students for an hour. You hand out last semester's homework. You go back to your real job. Anything beyond that, you hire someone for near-minimum-wage to do for you -- a graduate student who doesn't want to be there, or an instructional designer who can barely tie their own shoelaces (note: there are some really good IDs; just not at most universities. No one qualified would work at that salary or position in the social pecking order).

Universities which have strongly invested in online, blended, virtual, and pedagogy are surging, though. Why would I pay a $40k tuition when I can get GA Tech, ASU, SNHU, or what not which give better quality education for a fraction of the cost?

You will need the knowledge which the Bachelor would have taught you. If you've already got that knowledge through other means, to the level that a PhD requires, then actually getting that degree should be pretty trivial. Do it as distance learning, and treat it as a refresher to make sure you're as sharp as you think you are.
Is that really true though? My school offered the equivalent of CS 101 to CS grad students, so presumably some came from other disciplines and lacked the fundamentals.
Don't listen to anyone here. Just contract some advisors that interest you and see what happens and don't get discouraged. Don't waste your time going through the application process until you have developed a relationship with an advisor and they'd like to have you on. Good luck.
I'm a professor at a PhD-granting math department. I wish I could endorse this comment, but unfortunately it's incorrect.

In general I'm not available to "develop a relationship" with people outside the system, and with no apparent credentials. I have too many demands on my time, so I prioritize developing relationships with students enrolled at my university.

The one exception is if someone were to bring a genuinely useful idea to my attention. If someone had a good idea for improving the error terms in the Davenport-Heilbronn theorem, clearly understood the related literature, and brought this to my attention -- then I'd cheerfully work with them, even if they were an unemployed high-school dropout. But in practice this does not happen.

I also am happy to answer questions about our graduate program, or my discipline, from strangers. For example if a stranger wrote and asked "How can I best learn analytic number theory" then I would answer them. But for people who seek out some sort of advisor relationship, then all I can really do is to encourage them to apply to our PhD program.

> If someone had a good idea for improving the error terms in the Davenport-Heilbronn theorem, clearly understood the related literature, and brought this to my attention -- then I'd cheerfully work with them, even if they were an unemployed high-school dropout. But in practice this does not happen.

Yes, it does. That's the exact approach I used to get into grad school, and what I encourage others to do. Find a professor you like. Read their literature and get up-to-speed on their field. Engage with them around their work.

It's rare for people to do this, but it works. Sometimes you might need to do this with 2-3 professors to find one who is hiring grad students that semester (and be careful about programs where one professor can't hire but students are brought into a program), but it's a much better approach than spending a decade optimizing to GPAs, GREs, and what-not.

When someone does this, professors can have complete confidence the candidate can do the work, since they already are...

Can I ask what field you're in?

In mathematics in the US, "hiring grad students that semester" is not a common thing, and "programs where one professor can't hire but students are brought into a program" describes every program of which I'm aware.

But, I stand corrected that my experience doesn't generalize as extensively as I had imagined.

I'm in electronic engineering, but most of the experience is from computer science departments.

There are programs like Caltech where students do rotations and aren't admitted into a lab. But for example, at the MIT Media Lab, a professor can explicitly pull in a student for their group (and it's bloody hard to switch, if there's an advisor-advisee conflict).

And some departments are a hybrid, where a professor can pull someone in by writing an appropriate recommendation note to the admissions committee. If a professor says in a rec "I've worked with Alice for the past few months. If you admit Alice, we have a project defined together, and I have funding to pay for it," that's more-or-less an automatic admit, barring exceptional circumstances.

Interesting.

My guess is that in EE and laboratory-based CS, professors have a lot of work that they can readily farm out to people working under their supervision. This is less true in math, where it typically requires a big effort to get a student up to speed.

Sorry to be a jerk, but quickly rule out professors like this. Go listen to Maria Konnikova's interview on the Longform podcast here https://longform.org/posts/longform-podcast-400-maria-konnik... about her working with an advisor that was skeptical about the academic system. It's largely a racket and I say that having gone to an Ivy League and having worked in an elite lab in high school before working for Google. I would also listen to Thiel's talk on YouTube "Competition is for Losers" -- about creating genuine value and side-stepping, to the greatest degree possible, competition on metrics that a lot of people care about but just don't matter.

I want to appreciate the academic system, but unfortunately so much of it is prestige and credential games. It's best to step outside of that and work with others that are beneficiaries of these systems as opposed to slaves to them. It took me years of finding a degree of independent success in spite of hating the Ivy League I went to (thinking it was my fault for years) to get over this mentality.

If the last 5 years of working hard as an IC at a startup that had a 1B+ acquisition -- and watching others, too -- it's that you need to (a) be dedicated (b) work hard (c) work with others that believe in you (and only care so much about credentials) to accomplish great things.

So many are precluded from accomplishing great things due to a sense of miserliness and scarcity that academic systems create (avoid individuals with this mentality/do not cultivate this mentality within yourself at all costs).

There are wonderful, lovely things going on in academia, but also unconscious yet incredible MLM/Ponzi scheme-like things, too. Chose your path carefully and again, don't get discouraged.

That said, perhaps academia isn't right for you, too. And don't get discouraged by that either.

Based on the comments in this thread.

Could someone buy a way into a PhD? Provide funding with a requirement 'x' person must lead or be part of or funding will dryup?

Money opens some doors, yes, but this will be large sums of money. Not something like 1 000 dollars. Add two or three zeros. This also won't pass unnoted. If somebody in the future wants to find out how the PhD was achieved, he/she will find out.
There's lots of honorary doctorates to people who contribute to the uni, or just contribute to the world in general. Usually people don't use the title but some do.
I don't know the system in the US very well, but here in Germany it is already very hard to start an PhD as a bachelor student. One would need high/perfect grades and a supervisor which already supports him/her.

A PhD is not just learning something about theory but also applying theory and creating some new theory. I highly recommend you to start a theory oriented bachelor.

Hello! Math professor in the US here. Maybe, but probably not.

What might be easier to do is to audit a university course. Do you live in a city with a decent university? Find a course that interests you, and for which you think your background would be sufficient. Usually university course schedules are open to the public, and individual professors are easily Googleable. Professors who have personal webpages with lots of recently "cool stuff" tend to be better teachers on average, and also maybe more approachable. (This is, of course, a vague trend, not an ironclad rule.)

Anyway, it's okay to cold-email a professor and ask for permission to audit their course. Keep it brief; introduce yourself, say a sentence or two about your background, a sentence or two about why you find their course interesting. Be prepared to move on to someone else if they say no or don't respond.

If you end up excelling in the course, then this would might further doors for you; the professor might be willing to give you advice as to next steps, or to introduce you to other people who could help you.

Good luck to you!

How might that be affected with the recent quarantine orders? Do you know if it’s possible to audit a “zoom class”?
Good question. Hard to predict. With in-person classes it would probably be more difficult now, as universities are much more stringent about maximum enrollments and much more concerned about managing risk.

With Zoom classes, I doubt it would make too much difference. Professors are perhaps more concerned about their workload, or about maintaining some kind of class "community", and so might be somewhat more likely to say no. But I still think it's worth asking.

If I was a betting man I would say it will not be possible. The odds of universities publishing zoom links in light of the zoom bombings[0] seems pretty low to me. Most likely students will be contacted directly by their professors/administration with the link(s) for their classes.

[0]https://en.wikipedia.org/wiki/Zoombombing

The odds of universities publishing zoom links are essentially zero. You might find individual faculty posting syllabi with Zoom links somewhere online that anyone can see -- this would likely fly under the radar.

To do this, you would definitely want to contact the professor in advance. Email addresses for individual faculty are in general published and very easy to find.

Stephen Wolfram (https://en.wikipedia.org/wiki/Stephen_Wolfram) dropped out of high school and college before getting his PhD.

All it takes is being someone who

* Writes three books on particle physics by the age of 14

* Gets into, then drops out of Eton because of boredom. He instead ...

* Gets into, then drops out of Oxford because of boredom. He instead ...

* Gets into graduate school at Caltech

* Receive a PhD

* Do all of the above by the age of 20

See? Easy!

Despite my sarcasm above, the above shouldn't be (entirely) taken as discouragement. A one-in-a-century genius like Wolfram can do the above before turning 21. It's not impossible that you are the, say, one-in-a-decade genius who can, without being so young, as impendia said impress a professor with your insight. That professor might be able to, in turn, see whether his department's graduate program can admit someone on a non-degree basis (not unusual at all), taking a class or two at a time for a grade but not in a formal program.

If you're not "good enough", you'll likely be able to apply the classes toward a bachelor's degree. If you impress the right people in the right way and if the university does not formally restrict admission to a graduate program to bachelor's degree holders, it's not impossible that you could be formally admitted. Or perhaps a professor will be able to recommend you to a program at a university that does not have such a restriction. There are a lot of "if"s, but it's also not completely impossible.

PS - Yale Law School admitted someone without a bachelor's degree a while back. He was a very successful corporate executive who late in life decided to go to law school and did very well on the LSAT.

Yes. Or at least it was in 1981. tl;dr I was accepted into two Ivy League CS Ph.D. graduate programs with them knowing I wouldn't have a bachelor's when I got there and wouldn't get one.

Longer version. At a "public Ivy" university, I had finished all the requirements for a CS B.S. in three years...except I needed one full semester's course load in, well, anything. Number of hours was all that was between me and a Bachelor's. I checked with a fair number of schools that had "Must have Bachelor's degree or reasonable equivalent" in their catalog, explained my situation, and said I had three professors who knew my situation and would write recommendations for me. Most said "No, but you sound interesting. Try us next year if you don't pull this off". Two Ivies and a Canadian university said I could apply. The Ivies both accepted me and I withdrew my application from the Canadian one before they decided.

Didn't manage to pull off the Ph.D. though; my inexperience at 19 had me not realize it's important to find out about personalities and reputations among faculty for grad school, and it turned out my advisor/dept. chair liked to flush a majority of his students (I was the fifth of my entrance year). It turned out to be relatively simpler to get my graduate hours transferred down to my undergrad school and pick up the Bachelor's that way and then get admitted elsewhere.

Things have changed significantly for graduate CS admissions at good schools in the last 40 years though. Generally you're now expected to have already done some research, and you should have a really positive recommendation from whatever faculty you did that research with/under. I suspect your best approach would be to first get a "professional Master's", but since those usually don't involve research, also get on a research project while there and impress the professor(s) running it.

I got an answer and a non-answer for you.

The answer is in line with most of this thread: possible, yes, probable, no. All cases I know are from people who made enormous contributions to their respective fields before they could enrol in a PhD program.

Most important is my non answer: Why do you want to do a PhD? Just because you want to learn more theoretical compsci? Do you fully understand what a PhD program entails?

I'm not trying to discourage you here, I want you to understand what you would get yourself into before you spend a lot of time, effort and energy, and possibly months/years of your life into something that won't fulfil your life interests.

> All cases I know are from people who made enormous contributions to their respective fields before they could enrol in a PhD program.

This is highly dependent on the field and the particular group you work with. Most PhD students I know came straight from bachelor’s and did maybe a year of research in undergrad but with no significant contributions, some did actual research with a group, some have a paper or two published. But yeah, it’s pretty much impossible to get into a PhD program without a bachelor’s degree.

I read that as "All cases I know [of Ph.D. students who did not complete their bachelors] are...".
Ah, that makes a lot more sense.
It is, but good luck finding yourself in that situation. I received an offer during my bachelors which was not contingent on me completing the degree, but it came after a helluva unusual story that I can't repeat here.

Display exceptional creativity and promise in the field in which you're looking to study, attract a lot of attention for it, and then maybe, just maybe.

I would strongly advice against it.

It seems like you are looking at a PhD like "the diploma to have" but not realizing what a PhD is in practice. Read a lot of what a PhD consist in: At least, expect to be lonely and work a whole lot of hours by yourself without much guidance or external motivation, you are becoming the expert during a PhD, so you will not find a lot of people that can help you when you have a problem. (I don't say that good and knowledgeable advisors do not exist! but they also have a lot to do in 24h a day!)

IMO, nothing prevent you from studying after your work like if you are already a PhD student, give it a try: Take any advanced CS theory book or better, a Math book in your area of interest and work it out until you managed to do most of the advanced exercises. If you think this is not a healthy/sane life/work balance.. For a lot of people a PhD is not either. Take your time and be sure to understand everything in the book at its full extent. While you are doing this or after having done a couple of books, take high quality research papers in your field (lots of them are free), and try to understand them, implement them and try to reproduce their results. If you want a PhD in Deep Learning, do not expect that the PhD will consist in implementing a network, testing it, and start again. You may need to understand a lot of information theory, statistics, inference, math theory, etc. Also, for a lot of research level interesting ideas: implementing them is not possible in current frameworks, Expect to make a lot of coding. In any other research areas, usually (advanced) math is still a requirement: P/=NP kind of stuff in computability, graph theory, formal (type) logic (eg look at the rules for the Julia subtyping system[3] pg. 12), game theory (AI), category theory (for functional PhDs), stochastic calculus for finance, etc.. You don't need to know it all, only the part that are relevant to your interest, but still!

This may take month with long days to finish. but..

If you like this experience, you have two choices: 1) You feel that you are ready to do your research on your own, and in that case you don't need a PhD as you can continue to advance by yourself by taking any resources available. 2) You really need the diploma for whatever reason, give a shot at online universities, or try to find fast-tracks to obtain a master degree. Finally find an advisor that is willing to follow you and find funds for your PhD position.

In any case, after the PhD acceptance is where all the difficulties start: You need to study a lot, learn to master rapidly new material even if it is not the material you like, experiment a lot, report results, publish in high impact journals, review papers, read papers, prepare presentations, give presentations, give lectures, give exercise sessions, find fundings, write proposals, manage students, manage several urgent overlapping deadlines, prepare datasets, long hours, poor salary, etc. Don't expect that the PhD is related in any way to a bachelor/master program in my experience.

And don't think this will never apply to you if you find the right advisor, several studies study the link between a PhD and mental illnesses (~30%) [1][2], this is not exactly related to your capacity to work hard but more to the capacity of managing so many sources of anxiety while still being productive and often lonely.

All in all, this can be an extremely amazing experience or the worst experience in your life depending of your personality and ability to manage your time! I didn't write to discourage you to do it, but try in baby steps what it is like to be a PhD and after that decide for yourself. If you are convinced, find good advisors that are willing to follow you or guide you to a fast track that can suit you before applying!

[1] https://www.sciencedirect.com/science&#...

> I don’t have much interest in completing my gen-Ed’s, but I genuinely would like to learn more theoretical math/compsci, and I can only get so far buying used textbooks on eBay.

Most people in academia are pretty big on publishing so it's not terribly hard to find out what everybody is working on. The notion that doing a PhD program will get you access to a lot of otherwise unavailable knowledge is for the most part false. You may just get it a little earlier for your area of specialization.

You can publish and collaborate with others without a PhD or institutional backing too.

If it were to happen, I would expect the requirements to be similar to what's typical for graduate programs that accept people with an unrelated undergraduate degree. Basically, on top of the standard graduate level courses, the individual student takes all the usual undergraduate courses or their equivalent while enrolled as a graduate student.

For random advice from the internet I'd say that getting a bachelor's degree via a conventional route will save you vast amounts of heartache and pain relative to trying to convince a hierarchy of gatekeepers to give you a pass through all the academic and bureaucratic gates you want to skip. There is a 0.0001 probability of passing through ten gates of 50:50 odds and none of the gates you will need to pass through are likely to be 50:50 and the number of gatekeepers is likely to be more than ten.

Or to put it another way, you are describing a goal that is almost entirely out of your control, almost certain to be unachievable, and where persistence is likely to irritate the people who you would need as allies for it to succeed. Though it is logically possible to get a Phd without a bachelors, it is an eminently unreasonable expectation.

The steps to getting a bachelor's degree are enumerable and the there are institutional structures in place to facilitate doing so...and if you start today in a few years you can make one happen. To me, that's a happier plan than one completely outside your control.

In the meantime, you could work your way through Knuth without running out of things to learn from a book. Good luck.

CS Professor here: Depending on the institution, it's possible to get a PhD without an undergrad or meet alternative requirements to get an undergrad along the way.

The trick is you need to get admitted to the program. This is harder but doable. To get into a very good PhD program, you typically need some demonstrable track record of research and/or a letter of recommendation from a professor vouching for your interest/ability/potential. Of course, others do get in just on transcript and test scores, but it's rarer.

You could get a job as a programmer in a research lab and then leverage the relationships from there. Or you could try working with a professor informally (perhaps starting with software engineering for some project) and go from there. Once someone knows you and knows you have interest and aptitude, the lack of an undergrad degree should be solvable for at least some universities.

Finally two pieces of advice: Ph.D.s are supposed to be paid. Not that well (think 25 to 30k a year for a low cost of living area, 45k for say NYC), but if not it's a major red flag. Second, the quality and trust you have in your PhD advisor is way more important than the institution. A Ph.D. is almost an apprenticeship under someone. It goes badly if that person treats you badly

I did this. with this specific goal in mind I started reaching out to professors working in areas I thought were interesting. The only reason I was able to engage with one of them were:

  - had an SV startup background and the professor was interested in spinning them off
  - someone who was friendly with the department and had a sterling reputation went out of their way to pave the way for me
  - had some publications in the field
you're absolutely right about 'some universities'. a couple CS departments were interested, but there was just no way the graduate school would let them bring me on.

still, it was a process. I worked as a paid researcher for a while, and was able to take grad classes as part of the (very generous) professional staff education policy.

with a good (graduate) gpa, and a good GRE score (coming at the GRE from a graduate perspective made it a lot easier), I was accepted at that institution.

sadly I never developed a good relationship with an advisor and wandered off after teaching undergrads for a couple years.

but yes, its possible if you care enough

(edit: thinking about it..if you do care that much, maybe its not that hard to just get a BS? or diploma-mill MS?)

(edit again: I'm a systems guy, and I strongly suspect that the theory people are very different - I think if you had a really substantial publication history that might be enough?)

> try working with a professor informally (perhaps starting with software engineering for some project) and go from there.

I’ve been wanting to do this for years. Any tips on how to find a project and a willing professor? I’ve emailed interesting professors but I’ve never heard back.

(I was even curious to start a website to match up researchers with volunteer programmers.)

Its hard, we get a lot of emails and ignore most of them. You need to find some interesting insight or question, not "can you work with me." Making me think about research is a welcome thing. Making me think about managing people, not so much.

Couple of tricks: try not engaging via email. Twitter works (but again, make it a stimulating conversation, not a "work work" one). Or if they have code on github under active development, try contributing there.

Thanks. I like those ideas. I’ll give it a try.
I'm not sure about the US system, but here in Norway you'd probably have a hard time getting a PhD without a masters degree - but you can get a masters degree without a bachelor degree.

I'm not sure why you'd want to try for a PhD directly - but it might be possible to get into an MSc program?

In the US you typically pay for masters programs but not for PhD. (And most if not all PhD programs don't require a masters degree. Sometimes a masters degree is given during the PhD, depending on the school).
Oh, so it's typical to go from bachelor's to either master's or PhD, then?

Sounds a bit odd from a Norwegian perspective - our former (German modelled) cand.mag + cand.scient was a bit like integrated master's + PhD - although we also had a PhD which was more like a post doc (depending a bit on subject - a medical doctor had a pretty "conventional" path without much research). The old cand.scient. was 2 years of mostly research.

We do, of course, generally not pay for college over here, so it's a bit difficult to compare in terms of "cost" - but PhD is generally a salaried position.

These days it's normally bachelor's > master's > PhD - which are 3.5, 2 and 2 years respectively iirc.

Ed: it's not uncommon for an employer to fund a master's degree though - especially one based on experience rather than a straight bachelor degree. Eg you might write a compiler for a custom language, or do some machine learning or ux research as a masters while an employer pay for the work.

Yes, the European model of master's + shorter PhD is different from the US model of a longer PhD. There might be exceptions in less sciency disciplines (e.g. social work or public policy).

And yeah, PhD programs in the US are typically funded (no tuition, stipend usually enough to cover cost of living) while Master's programs usually are considered cash cows for universities. Of course there are indeed people who get a Master's and then a PhD (sometimes because they're not sure they want to do a PhD, sometimes because getting a Master's might help someone get into a better PhD program than they would have otherwise).

yeah. traditionally in the US the top CS programs really only want to recruit candidates who were in for the full package. MS was really kind of consolation prize

that started to shift about 20 years ago with professional night classes for MS students (I think modeled after the night school MBA programs). this is just a huge cash cow, and the programs aren't really very rigorous - you're not doing research - its more like undergrad++

On a slight tangent, what if you dropped out similarly, went back some years later, improved your grades at the end, but ended up with an undergraduate GPA of slightly < 3.0? What challenges/possibilities would there be in getting into a masters or PhD program after over a decade working? Assuming for the sake of discussion a good GRE score.
Mortimer Adler (1902-2001) recounted in his (first) autobiography:

Nonattendance resulted in a series of F’s on my record. At the end of my senior year in 1923, after I had already been awarded a Phi Beta Kappa key and had paid twenty dollars for my diploma, I received a note from Dean Hawkes saying that I might attend the commencement exercises but that I would not get my bachelor’s degree because I had neither passed my swimming test nor fulfilled the physical education requirement for graduation. Having earned 135 points of credit (120 points sufficed for graduation), I was, however, permitted to enter the graduate school without a B.A. degree. Six years later, without having bothered to stop for an M.A. on the way, I received a Ph.D.—I say “received” rather than “earned” because the doctorate fell on me in spite of myself and what I did or failed to do. How that happened is a story to be reserved until later, but that it happened gives me the rare distinction, I believe, of being possibly the only Ph.D. in the country without a master’s degree, a bachelor’s degree, or even a high school diploma.

Aside: Swimming test? Apparently he went to Columbia, I wonder why they would have such a requirement.
Before Air travel became the norm, people used to travel by water, and unless you know how to swim, you may drown for whatever reason.

The educational institutions didn't want all that effort wasted so they started requiring swimming education for people.

Or something like that, don't quote me haha

I almost didn’t get my MIT degree because I had neglected to take the swimming test, though I’d been swimming since childhood. I finally casually went in to take it and was told “you’re just under the wire, we’re closing it tomorrow for renovations”.

I don’t know if my leg was being pulled or not as I was already working full time and just finished my degree out of principle (and good thing I did!)

Barry Mazur, who is a professor of mathematics at Harvard, also only has a PhD. He left the Bronx High School of Science in his junior year to attend MIT without a diploma. He then neglected to complete an ROTC requirement at MIT, but he was already accepted to Princeton's PhD program, so he never bothered to get his Bachelor's degree. A few years later, he got his PhD in mathematics at the age of 22, which remains his only degree.
I do feel the need to add that these examples of people who were too brilliant to need degrees are not very useful answers for the OP, as amusing as the anecdotes may be.
This conversation took a big leap from "I want to learn more", into "PhD". If you just want to learn more, forget grad school and go audit the courses you want. There is no need to go through the hassle of any degree if all you want to do is learn, and don't actually have a goal of getting an advanced degree.
Yes! You could approach a Professor at a public university here in the U.S. (like the University of Colorado at Denver in Denver, Colorado) who mentors PhD candidates and explain the kind of on-the-job experience you have relevant to the industry in lieu of a Bachelors degree and request a formal recommendation letter to be included in your application to the PhD program. You will need between 10-20 years of professional experience, preferably team leadership. Although I heard most people are rejected anyways for things that can't be covered on-the-job like knowing how to write a proper academic research paper.

Edit: I just remembered that the measure of success in addition to writing a thesis is being able to explain your argument in as much detail as you can in front of panel of experts while answering their questions for at least a couple of hours up to several hours. Something you will likely encounter trying to convince a mentor you're ready for the PhD program.

In the US, yes, one of my friends did it. He was already working in a lab during high school and had published papers with them before he applied. They were first author at the top conference and he was ridiculously smart and focused on his work. But it's definitely possible. Bit different from your case but should give you some hope.
Another, lower-risk option: at some universities, it's quite possible for undergraduates to take graduate classes, especially if they've had the relevant background. So you could try applying to be an undergrad at such an institution. And you might not need to take that many gen ed. requirements if you find a university with flexible requirements (e.g., google 'open curriculum')

This is likely to be the easiest way to learn more theoretical math-and-CS subjects, though it'll probably be more expensive than just getting into a phd program.

I have no interest in the gen-ed bits myself, but you might take a look at the Coursera/University of London CS degree program. It's a 3 year program if you do it full time and it's entirely CS content.
CS PhDs are normally 4-5 years long. I powered through bachelors to PhD in 7 years (the short duration mostly motivated on keeping costs down). What you are describing is possible. I'm not convinced it is the best idea. A strong CS university undergrad may be 95% redundant, but that 5% matters. I expect your colleagues with degrees might have gotten less from their educations then you might in the same classrooms. We teach a lot more than people actually seem to remember. Don't discount going back for the bachelors. It might be worth it. A lot of your classwork for your PhD will be forcing you to take many of the same classes anyway.