[Danny, I composed the first part of this post before I saw your last comment. I need to rewrite it, but I won’t have the time today. The general arc of the argument is the same, though, so I’m going to post it as is.]
The ill-defined “rigour of mind” is something all mathematicians must have but not all people who have it must be mathematicians … so if the number of classes or formal qualifications was the measure then you and I would be more of a mathematician than a Fields Medalist!! [referring to Ed Witten]
Sigh. You have been tip-toeing around my requests because you don’t recognize why it’s an important issue, but you simply must address this: What is a mathematician? He cannot be ‘one with a rigorous mind’ (i.e., more capable of being correct) because others who have rigorous minds are not mathematicians, and there are plenty of people who call themselves mathematicians who are not the least bit rigorous. I know it sounds puerile, but I simply must know at what point this intellectual prowess begins, which particular mathematical concepts (and their application) one must understand to qualify as a mathematician (or physicist), and what it is precisely about these concepts that allow one to definitively intuit the limits of what can be measured or inferred (i.e., where science ends and informed speculation — philosophy — begins), one of the significant themes of this thread. In practical terms, I also want to identify the point at which I can ensure my own personal intellectual unassailability, thus freeing me up for truly informed inquiry, so I need to know if I can master these subjects, or if I’m already there.
You see, I believe that that we will in very short order start to reach the limits not just of what is measurable (observable), but of what can be inferred through mathematics. In an era of industrialized discovery, of commoditized innovation (I work in Silicon Valley), I think there are a lot of people making the very basic error of thinking that the pace of discovery over the last few centuries will continue in perpetuity. But, really, how likely is it that a bunch of monkeys (or genetically re-engineered monkey-derivatives) and their machines are capable of continuing this pace of mathematical innovation for 200 TRILLION years, when the last of the red dwarfs burns out and the universe finally goes dark. Alternatively, how likely is it that a bunch of monkeys needed only a few centuries of organized, industrialized innovation to come within a hair’s breath of describing reality down to its most irreducible layer?
Put another way: While some of us are greatly impressed with ourselves, in all probability our knowledge of the universe will never amount to much more than just a few thousand years worth of monkey business. In all probability, what we can never properly conceive and can never possibly understand will occupy us peripherally for an additional 200 trillion years beyond this frantic period of discovery, and all that time will be endlessly filled with philosophical speculation and scientific frustration and tantrum (hey, maybe they’re synonymous 
. In the end, and sooner rather than later, after we’ve run out of things we’re capable of describing or understanding, mathematically or otherwise, the ‘philosophers’, in whatever guise, will rule our affairs, not the scientists or the Higgs boson. I believe our destiny is a few thousands of years applying science to build our machines while re-engineering ourselves, followed effectively by an eternity of moral and spiritual speculation in near infinite variety. The problem isn’t the unknown, it’s how much of the unknown is unknowable. That formulation is trite sounding and the above explanation overwrought (after all, I’m not a philosopher, or a physicist , but try to take is seriously, because it really is the crux of the matter. You believe that the clever mathematical tricks of monkeys can infer all that is worth inferring; others believe the safer bet is that we’re outrageously inadequate to the task.
If I’m correct, I believe at least A LITTLE humility is in order, don’t you think? If I’m wrong, I simply must know which mathematical equations (and the physical concepts described by them) are required to disabuse me of this notion. Who could pass up on the opportunity not just to identify and understand all that is worth identifying and understanding, but to be all-knowing and all-understanding relative to one’s peers (i.e., a god), and all this just a few college courses away ….
