Mathematics and the Real

To my previous post, Louis Kauffman, himself, generously took the time to reply. I have included his reply in the comments section of that post. In that comment, I’d promised to continue the discussion in a new post. Here it is, with the brief continuation of the dialogue I had begun with Mr. Kauffman. I have copied the discussion here below:


June 15, 2015 at 4:49 am

Dear Lou,

Thank you so much for the time you took to answer. May I have your permission to copy your answer as a comment on my blog post? I will, naturally, link it to here.

Though I understand your point, I hope you can understand how dissatisfied I can be with such an approach: That mathematics is allowed such a ‘hand-wavey’ pass from reality – saying it’s just so because I said so – just seems like cheating. I guess, like Leibniz, I am more deeply convinced that _this_ is already an ideal world – necessarily-so, and so feel that our human attempts to break from reality does not address the real beauty of what mathematics could be (Thankfully, this finally ties-back into your own OP 🙂 ).

With Godel’s Incompleteness Theorems, we’ve seen that Set Theory is necessarily incomplete – for the sake of consistency. For some reason it seems that mathematicians have lauded consistency over completeness, where I think the opposite should be more praise-worthy (leave nothing un-proved unproven, and contend with inconsistency). You recognize the temporal nature of the Liar Paradox – this is just one ‘useful’ out from the problem, there are others too. We humans can be very creative when posed with such problems and it will push us towards a more ‘real’ mathematics. I personally feel that inconsistency is what imparts our reality with the very possibility of change – so gives rise to the very rich and varied world of mathematics, our language to describe change. That is why I haven’t allowed myself to ‘dictate’ that the distinction is continent. I guess I’m trying for a (IMO) ‘more honest’ theory of Forms…

I am not discouraged, and will drudge on. (Churchill’s quotation applies well here: “If you find yourself going through hell, keep going!”)

Once again, many many thanks and kind regards,

Tom


June 15, 2015 at 8:04 am

Dear Tom,
You can certainly use my comment or link to this blog. I remind you that I did not write Reality. I wrote “Reality”. From my point of view, all realities we know are relative to the way we make or perceive them. There may be an absolute reality in back of all of that, but in mathematics we explore various idealized realities that are are brought forth via our axiomatics and conceptual structures. These mathematical realities can be explored via logic and intuition. Many of them have deep historical roots, such as the natural numbers and classical geometry. Some of them are quite new and under investigation for the first time (by us!). This is the way it is for us.
Our perceptions of the world are not absolute. In fact, what we perceive in physics and other natural sciences are our own summaries and constructions of ‘what there is’. I do not know any way to carry on a mathematics that could be regarded as the ‘absolute truth’. Most of us mathematicians who were brought up in the 20th century regard this discarding of the notion of absolute truth in mathematics as a great liberation. We continue to believe in the power of human reason and keep looking to see what may come forth. In the same vein the incompleteness results tell us that specific formal systems cannot capture all of (relative) mathematical truth. This is to be celebrated. All of this is one important reason why I believe that it is very important to distinguish mathematics from the ‘physical world’. That way one can keep open the possibility that physical world shows more than we will ever find by pure thought and that conceptualization may be an enterprise that goes beyond the physics that we know now.
Best,
Lou


 

My opinion differs from Lou’s with regard to the ‘reality’ of Mathematics. Unfortunately I think, we’ve come to make a distinction between ‘Pure’ and ‘Applied’ mathematics – and yet I’m convinced that such a distinction should not carry with it the connotation that ‘pure’ mathematics is somehow ‘unreal’. We, human, conscious minds, may ‘create’ what is considered pure mathematics, but we are a product of this universe – we are ‘of’ this reality – and so, though this may seem like a ‘weak’ argument, mathematics, even ‘pure’ is still very real – only it is only ‘not applied (yet)’.

Similarly, the very foundations of mathematics – the theory of ‘sets’ – are conceptual anchors for very real things: collections of things. It is meant to reflect our means of thinking at its purest essence.

Now, I also think that ‘sets’ are a mistake – especially in the various mathematical attempts to make it ‘consistent’ at the sake of leaving it ‘incomplete’. That, I feel, is an ethical wrong. Science prides itself on proof and scepticism, and yet it uses a conceptual tool which accepts to have within it ‘unproven’ statements?!? I find that quite hypocritical. Now, that’s not to say science is hypocritical – it is to say that science doesn’t know that it’s being hypocritical.

While Lou ‘celebrates’ incompleteness, I do not. I think if we’re to question everything, and investigate reality via the scientific method, then we should hold our mathematics to the same standards: What is not falsifiable is not ‘science’ or at the very least not ‘interesting’ and I think the same of Mathematics. So a mathematical foundation that does not, ab initio, recognize this, is not ‘interesting’. Why is this? Because ‘pure truth’ (without the limit of falsity) tells us absolutely nothing. Like a newborn child for whom ‘it’s all true‘, they know nothing. It’s not for nothing that one of the first words we teach newborns is “No” – we rein-in a child, show and teach it limitations. Pure and absolute sameness is Nothingness. Without distinctions, without difference, we have no thing. So, for example, from my perspective, “Truth” is worthless, it is “Nothing”. It’s “Falsity” I’m seeking. I am trying to discover boundaries. Just like in science: Truth is only true until it’s not. All scientific ‘laws’ are only ‘laws’ until a better one comes along. And a better ‘law’ will be ‘more true’ because its boundaries encompass the ‘outliers’ of the previous laws, or perhaps ‘refine’ (i.e. draw clearer distinctions) our understanding of an older theory.

To me, it’s all truths-with-distinctions:

TRUE|TRUE

We perceive this as

TRUE|FALSE and vice versa FALSE|TRUE

So Truth is the ‘space in-between’, it’s the ‘normal state’, the ‘hum-drum’ boring thing. What’s interesting, even vital to me is the ‘interstices’ of truth – where things are no longer true – so that we can ‘draw a clearer shape’ of reality.

 

And that’s what gave rise to my ‘adjacency maps’… I am looking at George Spencer Brown’s work through a mirror. All my ideas are mirror-opposites of his. I find that bizarre, but I will explore these ‘adjacency maps’ further, in a later post. Keep your eyes peeled, coz it’s gonna be intriguing, and a real head-scratcher. And I’m gonna need some help…

Thank you for reading 🙂

Tom

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