Why should I enter a pure math PhD program if I most likely won't end up in academia? Assumptions:

The typical pure math PhD-bound undergraduate goes into a pure math PhD because they are [insert reasonable prerequisites for pure math PhD, (not the focus of this question)] and they want to end up in academia.
(With some exceptions) A pure math PhD is for people who want to enter academia.

Since most pure math PhD students won't end up in academia, we must wonder, given the above assumptions, the following question.
Question

Aside from the easy answer (see below), why should the typical pure math PhD-bound undergraduate enter a pure math PhD program if they will (1) probably not achieve their goal of entering academia and (2) probably have a few hard years after their PhD in chaotically bouncing around post-docs (never knowing when the end is in sight) and eventually readjusting their career goals to go into industry?

(1), (2) are based on observations one can easily make by browsing this site and talking to professors. This question may be abstracted to

Why should one go on a 5 year journey to achieve something they will probably not achieve, and upon failure, will subsequently face irreversible consequences (i.e., time lost, financial insecurity, etc.)?

The easy answer:
The typical answer is: &quot;Follow your heart, and do it if you can't imagine yourself doing something else.&quot; But this is an emotional answer which is automatically true for every undergraduate who is very serious about a pure math PhD. And just because you think something will make you happy doesn't mean you should always do it. Sure, a PhD is fun, but it seems like a PhD gets really not fun very quickly upon graduation.
Overall, it's very easy for naive undergraduates to accept this &quot;follow your heart&quot; answer because us pure math PhD-bound undergraduates tend to think &quot;Others have trouble, but I'm special.&quot; That's an extremely naive and unrealistic way to address the fact that most graduates from pure math PhD programs experience employment difficulties.
My background:
Mathematics undergraduate applying to PhD programs. I like math, I think I'm mentally and academically prepared to enroll in a pure math PhD program, and I have aimed to enter academia for a while. I know that with the right institution I would love a pure math PhD, but I want realistic career goals and not some naive expectation of &quot;grad school -&gt; PhD -&gt; post-doc -&gt; tenure&quot; in going into a pure math PhD.
Note: My background is not really a background. It is included to reinforce the fact that I'm your average pure math PhD-bound undergraduate, who would probably do just fine in PhD program, and therefore this question might apply to similar undergraduates who are also not trying to be so naive about their career goals.
why should I enter a pure math PhD program with the goal of entering academia if I will (1) probably not achieve this goal, (2) probably have a few hard years after my math PhD, and (3) probably end up in industry in which case I could have taken a less stressful route and even prepared myself better with a PhD in another field?

With the mindset that you are showing by the very loaded way your question is phrased - including several rather misguided beliefs I will address below - consider the possibility that the answer is “you shouldn’t”.
What are these beliefs I’m referring to? They are:

Something is not worth trying if the probability of success is lower than 50%

The definition of “success” in pursuing a pure math PhD is to have a career in academia

Any argument for pursuing a pure math PhD is not helpful or worth considering if it is “vague and subjective”.


In fact, these beliefs are not true in any universal sense. The decision to pursue a PhD in pure math is a personal and completely subjective one. It’s right for some people, and not right for others, and for each person it’s either right or not right for different reasons: e.g., for some people who do define success as an academic career it’s still the right decision because they are talented enough to defy the average not-so-great statistical odds you are referring to; others simply define success in completely different ways.
Likewise, the people for whom the decision does not seem like the right one also differ in the reasons why they reach that conclusion. Some are passionate about math but don’t think the effort, stress and opportunity cost are worth the risk of not achieving the dream of an academic career. Others may be so mathematically talented that they can know with a fairly high degree of certainty that they can have an academic career, and yet nonetheless find other opportunities more alluring.
Your question suggests that you want someone to come here and persuade you to do a pure math PhD by giving you some objective, scientific reason to do it. That’s not how it works. You’ll have to do your own thinking and make your own decision based on your own ideas of what matters most in life and what kind of life and career you want. I said at the beginning of the answer that it sounds like a math PhD may not be right for you, but of course I don’t know you and may well be wrong about that impression, so don’t let it prevent you from making the choice that’s right for you. Good luck!