One of the many pleasures in having my YouTube channel is getting to observe and participate in lots of spirited discussion by a wide range of viewers making comments on my videos. Here is my latest video in the MathFoundations series:

MathFoundations178: The law of (logical) honesty and the end of infinity

Even after one day, I have had many interesting comments. I would like to take the liberty of sharing with you two particularly cogent and insightful comments. The first is by *Karma Peny*, who writes (I have added some paragraph breaks):

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Excellent video; I could not agree more that it is time to expel “infinity” from mathematics. Not only do we need to define fundamental concepts with more clarity, but we need to define exactly what mathematics is. After thousands of years we still have no clear statement to describe what mathematics is.

In the early days of mathematics, all fundamental axioms were derived from real-world objects and actions. Any dispute over axioms could be resolved by examination of real-world objects and actions. As such, fundamental axioms were to some extent ‘provable’ by studying real-world objects and actions. Mathematics was devised to solve real-world problems and it was underpinned by real-world physics. Essentially mathematics provided a modelling tool to help us manage quantities of objects, determine measurements and to make predictions about the real-world, such as for engineering purposes and in astronomy. Many real-world scenarios have the same underlying physics, and so the same general-purpose mathematics can be applied to all cases. The addition of 6 apples to 2 apples has the same underlying mathematics as the addition of 6 pears to 2 pears.

This apparent generic nature can create the illusion that mathematics has its own ‘existence’ and that it is not simply a tool based on real-world physics. This will annoy many mathematicians, but the fact is that to claim that something is not related to the physical universe is to believe in the supernatural. This is what a belief in the supernatural means… the acceptance of phenomena that is not of this world. Whether maths is in the chemistry of the brain, on a computer, or in written form, it consists of rules devised by humans and all of maths has a physical presence.

To claim it has its own inherent existence or that it is in some way detached from reality is to turn maths into a belief system. The axiom ‘an infinite set exists’ is of equal value to an axiom that states ‘the god of thunder exists’. We can claim it is consistent and cannot be disproved, but both these axioms are equally worthless and irrelevant in the real-world, just as are any deductions derived using these axioms. It is often argued that the use of ‘infinity’ in mathematics has proven to be very successful, but the successes could be despite the use of ‘infinity’ rather than because of it. I suspect we will have more clarity and even more successes if we abandon the use non real-world axioms.

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And now here is the response to *Karma Peny’s* comment by *Amanojack A, *a consistent contributor of well written and insightful comments. (I have made a single spelling correction.)

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I think you have it exactly right. Math was born out of finding useful abstract objects and situations whose relations were isomorphic/homomorphic to various real-world situations. In other words, a mathematical field’s objects, “moving parts,” and those movements and relations usefully corresponded to certain objects, moving parts, and their movements and relations in the physical world. Pin down the math and now you have a powerful tool applicable to any real-world situation as long as it has an aspect with a homomorphic correspondent in the math. For example, pin down multiplication and you have a powerful tool for counting how many apples you have if they come in crates of 24 each.

So-called “pure math” was born out of the idea that it might be worth developing mathematical objects and relations that correspond to no physical situation yet discovered, but could. Seems noble enough. The problem came when people failed to keep track of context. They floundered into musing about things that not only had no known physical analog, but that **couldn’t ever even conceivably** have a physical analog. They were unpicturable, things we “only imagine that we can imagine,” as Wildberger said. Like infinity. In another comment I elaborate on how this mind trick is pulled off, making us think we can imagine something we really can’t.

When physicists objected, mathematicians like Hilbert decided to take over the physics departments as well – such has been the power of this social trick of intimidation by pretending to have a unique ability to imagine the nonsensical. Paradox thus became a badge of honor, a sign that you were approaching deep wisdom (rather than stumbling into incoherence). We live with results; they now affect every field, as people point to how physics – king of the sciences! – gets away with it. The infection started with math, spread to physics, and after a century has turned into an epidemic with tendrils extending even as far as the art world of all things.

Returning mathematics to a solid footing is of paramount importance to all fields, as math is the standards bearer for rigor. It does a good job with logical rigor but tends to ignore semantic rigor as is convenient, which in turn lets all other disciplines off the hook in this regard, weakening all of academia (physics being the main conduit).

You hit the nail on the head when you say the successes of mainstream math have come in spite of infinity rather than because of it. Just like the axiom, “There exists a god of thunder,” the axiom of infinite sets functions as a cultural license; it simply allows those figures with the most authority to make up whatever fudges they want to make it look like they’ve proven something rigorously when they haven’t. The resulting mathematical world and its engineering applications retain the appearance of being held up by mathematical rigor, but they are actually held up variously by fudges handed down by fiat and by engineers adjusting them to avoid the cases where they break down. In other words it’s a big mess that is shoehorned into a usable framework, but not by the rigor of mathematicians – that is just smoke and mirrors (see calculus, example; “we’ll prove it rigorously, with limits!” – no, we’ll just make a show of it and move on, knowing it already works well enough for engineering).

In a sense, then, infinity has been quite successful…as a tool for advancing people’s math careers and social standing.

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Thanks to both Karma Peny and Amonojack A for these penetrating comments!

Tom W.I’m glad to see the social impetus to revise mathematical foundations still beats underneath. Can we address the fundamental error of ‘membership’ in set theory too please? There is no null set just as there is no infinite set – zero and infinity are the same thing to the human mind – concepts of ‘sameness’ and ‘lack of distinction’. As such then all existents must be, by definition unique. To be unique is to be by definition ‘not’ all other extant things (and all extant things are ‘not’ this thing also). This makes every extant thing ‘adjacent’ to every other, simultaneously. As such this means that no single thing can be ‘in’ any other thing, thus not ‘in’ another ‘set’. Membership is a case of mistaken identity with… identity. A new system of thought and its description is necessary. I have tried (http://taomath.org/2015/06/adjacent-existents-a-theory/) but it isn’t easy to do it alone.

gentzenI recently quoted your new logical principle, together with one of your nice huge numbers:

I hope I didn’t misrepresented your point of view too much.

If you don’t believe in infinity, then what will become of the infinite regress in Agrippa’s trilemma? But an infinite regress is problematic anyway, because sometimes you cannot know whether it wouldn’t end up in one of the other two cases, if you just continued discussing with your opponent long enought. And I was surprised when I realized that a Turing machine can be in a similar situation, when it tries to determine whether a given input is finite or not. In fact, a Turing machine is unable to distinguish an infinite (=never ending) input from a finite one, even so in the end it will know for each finite input that it was actually finite:

ishindenThese comments about math taking over physics very much remind me of the essay Death by Mathematics (http://milesmathis.com/death.html). Have you read it, Norm?

Tom HolroydI’ve read about many examples of “completion” such as when you add “points at infinity” to the plane to get the projective plane. Nobody complains about that type of infinity. It’s just what you get when you divide by zero, move along nothing to see here, in homogenous coordinates.

Infinity is a process. Still, you can “close” many systems easily. Fermat already showed how to create a ring of dual numbers using non-zero nilpotents (which can be constructed) that square to zero, and you can use this to prove all high-school theorems in calculus WITHOUT using limits, or infinity.

https://en.wikipedia.org/wiki/Dual_number

Smooth Infinitesimal Analysis, Synthetic Differential Geometry, etc. At some point you just redefine what a number is (EXACTLY like using homogenous coordinates in projective geometry). Epsilon is a thing you can write down. Epsilon squared is zero. Epsilon is not equal to zero. You have to use intuitionistic logic (aka constructive logic). It’s not imaginary (neither is the square root of minus one, I can show you a two by two matrix that squares to minus the identity.

Infinity is a process, and the continuum cannot be described in terms of discrete objects.

I’ve also seen that there are some attempts (Utimate L) to close up the cardinals. Perhaps there is no reason to go up the infinite hierarchy? Perhaps it can be closed?

Jim G.Hi Norman,

I thought I should mention this “real number” conundrum that came to mind, which must have been noted by others somewhere in the past, although I have not located it. Recalling statements like:

e^(i*pi)=1

It is asserted that even powers of odd numbers result in even numbers, for instance:

(-5)^2 = 25

But exponents don’t have to be integral. I imagine somewhere there is an authoritative answer to the question whether

(-5)^(2/6) = (-5)^(1/3)

I am assuming (ignorantly) the convention in math would be one of these two:

A) divide the square of -5 by its sixth power, giving a positive result

B) reduce 2/6 to lowest terms, at which point it becomes 1/3, yielding a negative result

Regardless, since “oddness” is only a property of whole numbers (and can be said to extend to negative integers via the form -2n+/-1), this raises a serious problem for so called “real” numbers. For instance, Is this statement true? :

(-e)^(i*pi) > 0

or is this one true?

(-e)^(i*pi) < 0

Of course, as we know, the convention in this instance would be to let a broom B, with infinitely many bristles, sweep the whole business under a carpet C, of infinite area.

Jim G.OK. I have horribly bungled it! Let me try that once again:

So (-5)^(1/3) is the cube root of -5. Odd roots of negative numbers would have some sense to them.

But the expression (-5)^(2/6) could present a problem, even though 2/6 and 1/3 are equivalent. If we square the -5 first and then take a sixth root, the result is positive, opposite that for the cube root. But a common convention that sorts out the order of operations could get us past such ambiguity.

However, for (-5)^e or (-5)^pi, we have got a very serious problem, since these exponents cannot be represented as a signed ratio of whole numbers, and it’s those whole numbers that would be exhibiting “oddness” or “evenness” needed for us to clearly determine the sign of the result. There seems to me to be a gradual descent into meaninglessness going on here. First the cube root of a non-cubic number, -5. Now a root to an exponent that allegedly never ends. One can make approximations but indeterminate sign seems to be insurmountable absent some magic.

Jim G.One other thing: this situation seems to be even worse. Even if one were to try to approximate the sign of this negative number raised to an irrational power, there are problems.

Firstly, an approximation to an irrational number is really just a series of rational numbers, not the irrational number itself (whatever that would mean). And we won’t ever get to the end of that series, because the value is not irrational. We just progressively lessen the breadth of the inaccuracy of the result.

Even if we grant some kind of equivalence between the approximating algorithm and the irrational, there are still lots and lots of algorithms we can choose from to generate approximating sequences to a given irrational (for instance, pi), and one will typically give different fractions at each iteration than another one will.

But if we ignore this still, and we select some algorithm to be canonical, and we then use it consider approximations to pi, some of fractions in our resulting sequence, even expressed in lowest terms, will vary in the evenness and oddness of their numerators. For instance, in base 10, 314/100 reduces to 157/50, which asks us to take an even (50th) root of an odd number. This means we are restricted in which number fields we can use to resolve the calculation.

Even in a prime number base system like, say, 7 or 11, the same problem will emerge whenever the numerator of a given approximation switches about from odd to even.