On the Nature of Mathematics

Hands up, I'm way outside my sphere of expertise here: I'm no mathematician, physicist or epistemologist. I know enough to be dangerous, but not enough to make any ground-breaking contributions. So why read any further? Well, I have something to say on this which has been worming around in my mind for years, sometimes peeping into my consciousness for a fleeting moment, before vanishing back to obscurity. But it's now well enough formed to describe as a starting point, if not with any degree of eloquence. And even if I haven't made any breakthrough, let me perhaps lay a cobble along a road which may be interesting to walk. Please be patient...

I have long thought about the nature of mathematics in relation to physics. The two disciplines are closely linked, particularly at the limits of today's advanced cosmology and particle physics. Many physicists and mathematicians have marvelled at the predictive power of mathematics, at the way that theories can be synthesized mathematically into structures which suggest discoveries to be made experimentally, if only to realise some perceived 'beauty' by completing an elegant mathematical structure. When lo and behold the discovery is made, theorists marvel at mathematics and its pre-eminence among the intellectual disciplines.

Some go further still: Roger Penrose holds mathematics to be the reality of nature, and if that wasn't enough, that mathematical concepts have a metaphysical existence in the universe independent of mathematicians, as if Pythagoras's Theorem was floating in the ether for the Greeks to discover and document.

I have long held mathematics to be a human construct which represents the world around us, and not some disembodied mystical entity. I have found every other metaphysical construct to evaporate under the harsh light of critical examination, and I have no patience to entertain disembodied equations emerging from the Big Bang! But I see a problem: my view of mathematics as a construct just doesn't fit with the predictive power mathematics has proved to wield. The discovery of the Higgs boson was a triumph of the predictive power of mathematics, and one which has not sufficiently been heralded in my view. So there must be something more to mathematics than just a toolbag of strategies for solving practical problems.

There is something transcendent about mathematics. If you know what a materialistic skeptic I am, you'll appreciate the enormity of that statement. Solving equations, as I do from time to time as an engineer, feels like refining truth - cancelling terms feels like spooning off the dross from the ever more pure and precious metal sought. The formal proofs of theorems are eternal - once proven they are never broken, and reveal their truth for eternity. There is some kind of magic in mathematics, but I just can't follow Penrose down his metaphysical road. That way lies madness!

I also have bags of humbug for the ancient Greek philosophers. Hemlock wasn't Socrates's only herbal vice: just what was he on when he came up with the Allegory of the Cave? So I'm more than slightly embarrassed that my resolution to the problem of mathematics has certain similarities to his shadows on a cave wall.

While listening to back issue podcasts of The Infinite Monkey Cage a few days ago, with Brian Cox perhaps stating as final that mathematics is truth, while Robin Ince teases him on multiple levels simultaneously, which you only realise are much more clever than at first appears some time later, a thought popped into my consciousness, and decided to hang around.

The thought was: "there is a structure of underlying truth to the universe which we hairless apes are not adapted to comprehend, but parts of which are projected onto our limited consciousness, and the shadows formed are what we call mathematics".

Sitting there, like a mischevous imp at the corner of my mind, that thought cast off other thoughts. I thought of the schematic map of the London Underground. When laid out geographically, the tube network is fiendishly complex. But the schematic representation just shows what we need to know to plan a route from A to B, and where to change lines. It's a functional representation of London, but it's not actually London. So if the underlying truth is like London, but we only have parts of a schematic tube map, there are limited things we can know about London, (er, I mean truth). We know schematically that the Jubilee line crosses the Circle line twice, and if we know that in real London it crosses at one point, (we have solid experimental evidence for one physical law), then we can infer from the rules of topology that it must cross somewhere else (and make a prediction to test experimentally), even if we've never been to Baker St.

There are truths which are so obvious to us that they seem pointless to express: like the number 2 is half of 4, and sits neatly between 1 and 3.  Perhaps if we were not adapted to life as apes, but as supreme logicians, Pythagoras's Theorem would be similarly trivial, and unworthy of a name. So perhaps there is no need for a disembodied metaphysical law of right triangles in the universe, right triangles just are the way they are. And it's not obvious to us because we don't have the right kind of minds to appreciate it, and have to construct formal proofs instead. These proofs seem so magical and powerful to us, that some of us think they have a special existence, but that's just an illusion born of our limited perception. And perhaps the behaviour of waves and particles, and spacetime, and the unity of forces, are all logically deducible, if only we could perceive the logic so clearly.

So we build pieces of a reality map through our reasoning and by our observations, and call these pieces laws and theorems. But these laws and theorems are our constructs, our inventions to account for the way the universe is, to steer our ape minds to conform for a moment to the truth of reality, while the universe just goes on being what it is without any need for such trivia.

On this view then, mathematics really is the projection of reality onto human consciousness. And as the contours of our consciousness change, so do the mathematical strategies we use. When I learned basic number theory as a child, I used abacuses to count-on and perform basic addition. My children were taught the number line, which is a different concept. So their mathematics will be different to mine, not because truth is different for us, but because their consciousness of number is different from mine.

What can this idea tell us we didn't know before? Well it does suggest that there may be limitations to what we can discover. In terms of the analogy, there may be areas of our consciousness which our cerebral topography keeps in mathematical shadow, corresponding to universal truths we can never comprehend. But who knows, if we can find where these conceptual gaps lie, perhaps mankind's perseverance at solving problems will find routes around these gaps, allowing us to solve theoretical and practical problems regardless. Quantum theory could be one of those gaps - we just do not have minds equipped to understand the world on such small scales, but we have mathematical strategies which allow us to skirt the edge of our blind spot and solve quantum mechanical problems anyway.  We've done rather well for ourselves, don't you think?

Fiat justitia ruat caelum

Dame Elizabeth Butler-Sloss is under fire for having taken the chair to inquire into allegations of historic establishment cover up of child abuse, when her brother stands accused of being party to such cover up. There is indeed a prima facie conflict of interest here, and I've been thinking this one over all day, (off and on, among other things). Now I think I have reached a personal conclusion.

What needs inquiring into, (among other important things, but this one’s pretty central), is this: we had people in positions of authority and responsibility who are alleged to have been conflicted when they became aware of allegations of wrongdoing, but failed to pursue those allegations because that may have disadvantaged them in some way. Perhaps politically, perhaps in their career aspirations, perhaps because it threatened to bring into disrepute a social group to which they belonged. We've accepted these people into positions of authority and responsibility with the unspoken bargain that they should in return exercise such powers responsibly, for the benefit of wider society and not for personal protection or advancement.  And we've been let down: by cash for questions; by the MPs expenses scandal; by the failure to hold Jimmy Savile to account; by disgraceful cover ups over Hillsborough; by the phone hacking scandal.

It seems to me that there will be no one in the establishment with the authority to inquire into this matter, who is not tainted in some way by association.  And despite the appearance of conflicted interests, that is not what we need. What is needed is someone who has the integrity to rigorously inquire into these allegations despite it being potentially damaging to the good name of a close family member, and not someone whose potential embarrassment remains concealed.

All commentators who seem well enough informed to judge her describe Dame EBS as possessing impeccable integrity and unparalleled qualifications to chair this inquiry.  And continuing to chair the inquiry would give her the opportunity to set an example to those pillars of the establishment and inquire into this matter as rigorously as they all perhaps should have done all those years ago, and damn the consequences.

I think she should proceed, and show them what integrity looks like, because it's a lesson many in the establishment need to learn.

Profile: Dame Elizabeth Butler-Sloss
Fiat justitia ruat caelum

LogMeIn - Can that let anyone in?

Until recently, I used a service called LogMeIn, which allows me to access other computers I've set up with specific accounts.  I used it primarily to enable me to provide technical support to family members.  It's an incredibly useful service, and until recently a limited service was available completely free of charge. Recently the free option was withdrawn so I stopped using it.

I used it twice for business purposes, one time setting up a specific email address so that a client could access a remote computer I had set up to configure a specific piece of equipment installed on another continent.

Then I began to receive spam email which had been sent to the login email address I used for business that one time.  I knew that the email address had not been published.  I knew that only I had access to the server receiving emails to the address used.  I knew that I had not used it for any other purpose, and that there was no benefit to it being used by anyone else, since only I could receive emails sent to it.  The email address used for spam HAD to have been released by LogMeIn.  So I emailed LogMeIn and explained, but I just received a generic denial, and left it at that.

Today I received an invoice for $826, purporting to be from LogMeIn, sent not to the exclusive login account previously sent spam, but to my business email address.  The attached invoice contained no data, so I was fairly sure it was a phishing attempt to defraud me.  Sure enough, one look at the LogMeIn Facebook page reveals other victims complaining of the same scam.  I forwarded the information to LogMeIn, along with my rationale for believing that it was attempted fraud.

Now I think about it, my alarm bells are ringing loudly.  LogMeIn has allowed my private data to fall into the hands of criminals, witness the exclusive login email address now used for spam.  That's irritating.  Now their customers are being targeted with attempted fraud.  That's alarming.  But how bad could this be?

I used LogMeIn to gain authorised access to computers, with the knowledge and trust of the owners of those computers.  But the means to do that, in the form of login credentials to the computers involved, is entrusted to LogMeIn.  And I now know that information entrusted to LogMeIn has fallen into the hands of criminals.  So how safe are the computers I have used with LogMeIn?  If someone with malicious or fraudulent intent can gain unauthorised access to those computers, they may have access to all sorts of information which could be harmful, such as login details to banking services, financial accounts, employee personnel records, medical details, intellectual property, state secrets,... the list is endless.

I have removed LogMeIn from all my computers, and family computers.  But this remains a concern.  If LogMeIn cannot secure client data, and they hold the keys to millions of computers worldwide, is it wise to entrust LogMeIn with those keys?  Think about it... it gets frightening if you do!

LogMeIn also provides services marketed as RemotelyAnywhere, join.me, Xively, Cubby and BoldChat.