This was one of many assertions @Prycejosh1987: made, that are utterly wrong. Many other species have intelligence, and live in complex societal groups, even trading favours, in chimpanzee societies for example, alpha males will hunt and then use meat to reinforce alliances with those it most closely allied to.
Like many apologists he talks utter nonsense, and then refuses to acknowledge reasonable rational objections to his claims. Humans are just one species of great ape, and it is astonishing how closely our behaviours and instincts are mirrored in other species of that taxonomical group. Theists deny such facts, as they want to believe we were created in the image of divinity, and it’s hard to square that claim with the facts, for example that we share almost 98% of our genes with chimpanzees, as it makes the claim appear risible.
This HAS been said about mathematics because there are certain axioms in math that are not proven but math based on those axioms accurately models the real world so they are assumed to be true. Such as the axiom of extensionality in set theory.
But math is not typically regarded as science (or philosophy, for that matter). It’s a formal system or tool which works through logical deduction rather than empirical observation.
That depends on the type of math. Applied mathematics and industrial computation: Yes, it is definitely science. Pure mathemathics: depends on the subject, whether it is developing techniques for solving science-inspired problems or pure, abstract excercises with no known applications other than doing mathematics for the sole purpose of doing mathematics. If the former, perhaps. If the latter, perhaps not.
An important fact to remember about axioms in mathematics, is that they are chosen deliberately to be as simple and self-evident as possible, and furthermore, they are restricted to extremely simple abstract objects.
In short, mathematicians want their axioms to be obvious. Furthermore, they are intended to be foundational - namely, that no simpler statement about the behaviour of the entities of interest can exist.
The case study of course, centres upon Euclid, who for an individual operating in 300 BCE or thereabouts, launched an absolute tour de force with the writing of his Elements. This was the first genuine formal axiomatic system in mathematics, centred upon geometry, and he based that system upon five proposed axioms,
Mathematicians working after Euclid discovered that he had indeed laboured diligently on this matter. They found that it was impossible to construct a consistent geometrical system that violated any one of the first four axioms. As a corollary, those four axioms are considered foundational to any system of geometry, not matter how far removed from the Euclidean original that system may be.
The trouble, of course, lay with the fifth axiom (what was termed historically “Euclid’s Fifth Postulate”). This is the axiom centred upon the behaviour of parallel lines, and this one occupied the minds of mathematicians for two whole millennia. Try as they might, they couldn’t demonstrate that violation of this axiom led to nonsense.
Two mathematicians stepped into the breach (three if you count Bolyai - see below). Who, independently, asked the question “What would a geometry violating Euclid’s parallel line axiom look like?”
With hindsight, this seems like a question that should have been asked a lot earlier. But mathematicians didn’t have the tools to investigate this before this moment, so that rather put a damper on earlier attempts to answer this question.
The first to do so was Nikolai Lobachevsky. Who decided to investigate if other geometries existed apart from those of Euclid, and in 1826, demonstrated that such a geometry was possible. This hyperbolic geometry was also alighted upon independently by János Bolyai, and the resulting geometry is referred to either as Lobachevskian geometry or Lobachevsky-Bolyai geometry as a result.
This geometry is interesting, because it introduces the idea of space possessing an intrinsic curvature. Basically, the coordinate axes, coordinate lines and coordinate surfaces in this geometry are curves, not straight lines, and the curvature is of such a nature that there exist what are known as saddle points in the space - points where the coordinate curves for one coordinate variable move in the opposite direction to the coordinate curves of another coordinate variable. The resulting structure looks like a horse saddle, hence the name.
Next, Bernhard Riemann stepped into the breach. And produced a geometry different from Euclid and Lobachevsky.
Once Riemann started attacking the problem, he discovered something very interesting. His new geometry turned out to be perfect for analysing the behaviour of spherical objects, and spaces with spherical curvature (differing from the hyperbolic curvature briefly mentioned above). It also led to the development of the concepts of differential geometry and manifolds, the latter concept enjoying a lot of use in modern cosmological physics. Indeed, Riemann’s idea of a manifold turned out to be an alternative means of producing Lobachevsky’s geometry as well. But I digress.
The point being stressed, with this fairly extensive examination of the example provided by Euclid and successors, is that in the realm of mathematics, the aim has been, increasingly, to base axioms upon the simplest possible entities, and define the simplest possible behaviours thereof. From which, hopefully, a consistent formal system for those entities will emerge.
It is at this point, that the contrast between the operation of mathematicians, and the operation of pedlars of rectally extracted apologetics, will become stark. Pedlars of apologetics seek to establish as “axioms” assertions that are positively overstuffed with conceptual bloat, are centred upon entities that are (and indeed, necessarily must be according to some of the apologetics in question) farcically complicated, and involve entire cascades of presuppositions to be in place beforehand - this latter feature on its own disqualifying the assertions in question as “axioms”. (See my remarks upon the term ‘foundational’ above).
This, of course, points once more to the manner in which pedlars of religious apologetics, routinely and egregiously abuse the products of rigorous fields of human endeavour. That they are backed into this corner of discoursive duplicity, arises directly from the nature of the assertions being peddled and desperately clung to. The assertions in question have no support from observational reality or logical deduction, so the pedlars of the assertions in question have to engage in discoursive malpractice. In the Internet age, one can describe said malpractice, with acidically barbed satirical fidelity, as bad science and philosophy cosplay.
Indeed, I would urge those repulsed by said malpractice, to deploy that satritical barb whenever it proves to be most critically damaging, and savour every last atom of discomfort elicited from the fraudsters in question, because at bottom, the sort of apologetics being considered here is not merely ridiculous, but is de facto fraudulent. Said fraudulent nature of the apologetics in question needs to be illuminated with the brightest of discoursive spotlights at every opportunity.
I suspect this covers relevant bases, but will of course welcome additions that I’ve missed.
Yes, rather like how software is designed. A minimum of entities, a maximum of descriptive power, is always the sweet spot.
I enjoy learning from your posts … and I learned a great deal from this one. TBH, as a builder of line-of-business software, I’ve never needed to go beyond high school algebra in practice. If I were building a game physics engine or an LLM, it would be a different story. At 68, I am probably not going to learn discrete math though. Although you never know, when I finally retire I may decide to use my free time in ways like that.
In mathematics, we can assume the truth of an idea, and then disprove this idea by working with it until we reach a contradiction.
The square root of two was proven to be an irrational number because we can work out a math problem that assumes that the square root of two is rational, and then show that this number is both odd and even at the same time . . . which proves that this number is irrational.
So, let us start by assuming that God exists.
If God is all powerful and has no limitations, then God can do anything He wants.
So . . . can God create a rock that is so heavy that it would be impossible for Him to move it?
