Hannah Arendt

 I watched a movie titled, Hannah Arendt (made in 2012), which was a joint European, Israel production, and is classified as ‘biography’ on SBS (where I saw it). It’s a dramatised biography, though it contains archival footage of Adolf Eichmann’s trial in Israel. The story centres on her coverage of the trial for The New Yorker and the controversy that followed. But there are also flashbacks to a younger Hannah, when she was Martin Heidegger’s favourite student and subsequent lover.

It so happens, Philosophy Now (Issue 143, April/May 2021) also had a reasonably in-depth article on her (pp. 50-3) by Hilarius Bogbinder, a Danish born writer and translator, who studied politics and theology at Oxford and now lives in London (according to a footnote).  I re-read the article after seeing the film, and they cover much the same territory.

 

In the movie, she’s portrayed by Barbara Sukowa (who appears equally fluent in German and English) as a feisty woman, capable of holding her own with any intellectual or authority figure. Given she spent time in German occupied France helping Jews escape, before escaping to America herself, one could imagine that she would not be easily intimidated or bullied. Bogbinder describes the young Hannah:

 

Bright and precocious, Hannah [originally Johanna, but always called Hanna] was a bit of a handful. A chain-smoking, self-confident young woman, in the 1920s, she stood out from the mostly male crowd at the University of Marburg.

 

In the movie, she has a sharp mind and a sharper tongue if pushed into a corner, which happened more than once, by various forces, following her coverage of Eichmann’s trial: including Mossad, Princeton University department heads, a particularly vicious journalist from The New York Times, American Rabbis and a large proportion of the New York Jewish community. However, her students loved her.

 

According to Bogbinder, she said in a TV interview in 1964, “I am sorry, but I have to object. I don’t belong to the circle of philosophers. My profession, if you can call it that, is that of political theory.” Yet she was the first woman to teach philosophy at Princeton University. The impression one gets in the movie is that she cared very much about thinking and what I’d call ‘authentic thinking’. To me, philosophy is about argument going back to Socrates and Plato (she references that lineage at one critical point in the movie) as a bulwark against dogma and political ideology. I’ve never read any of her works, but apparently one of her most influential works was The Origins of Totalitarianism (1951) where she wrote:

 

While people in all great revolutions fight for true representation, the mob will always shout for the ‘strong man’, the ‘great leader’. For the mob hates society from which it is excluded... Plebiscites, therefore, with which modern mob leaders have obtained such excellent results, are an old concept of politicians who rely upon the mob.

 

Amazingly prescient, if one looks at recent American politics. Bogbinder makes the following observation: “Yet it is not difficult to see how many of her acute observations transcend her times, and apply to the 2020s, when demagogues denounce ‘experts’ and the ‘elites’.” This is also particularly relevant to modern-day Australian politics (for the past decade at least).

 

In this context, one can see why she might have seen herself as a ‘political theorist’, as opposed to ‘belonging to a circle of philosophers’. In one dramatic scene (in the movie), in response to a jibe from a fellow professor in a classroom full of students, she says, ‘That’s not an argument; that’s character assassination’. Maybe the experience at Princeton, following her treatment by her colleagues, made her denounce philosophy as a discipline. She also severed intellectual links with Heidegger after he became a Nazi, which would have disillusioned her.

 

A.C. Grayling, in his ambitiously titled tome, The History of Philosophy, gives her one paragraph, where he says:

 

She is an exemplar of intellect that is both penetrating and courageous. Her chief concern was to argue for the importance of political engagement as a civic responsibility, in order to defeat totalitarianism.

 

According to Grayling, she forgave Heidegger after the war. I won’t try to get into her head on that issue, after all they were once intimate.

 

Even people who have never studied philosophy or who are only vaguely aware of the horrors of the holocaust, know the name of Hannah Arendt because of the term, ‘The Banality of Evil’, which was the subtitle of her book on Eichmann, a publication of the reportage she had submitted to The New Yorker in serial form. The movie provides context for this famous utterance, especially when archival footage shows us firsthand what Eichmann said in his defence. In effect, he was a bureaucrat who was following orders, and ‘obeying the law’ of his government. Hannah Arendt’s most damning indictment against Eichmann was that he didn’t think, in fact, refused to think. The point is that she refused to paint him as a demonic monster because he wasn’t: he was an ordinary person; in effect, a ‘non-person’.

 

But, not only was she accused of ‘defending Eichmann’, which is ridiculous, but of being a ‘self-hating Jewess’, because she reported that some Jewish leaders had been complicit in the transport of Jews. When questioned on this, she simply said that it came out in the trial, so she was obliged to report it. Many in the Jewish community never forgave her for this, and her works didn’t appear in Israel until 1999, when her report, Eichmann in Jerusalem (1963), was finally published in that language.

 

Very early in the history of this blog, I wrote a post titled Evil, in response to a book written by TIME Magazine essayist, Lance Morrow. What struck me was that Morrow seemed to believe that evil required a specific personality type. I’ve never believed this. History, including recent history, is full of stories of ordinary people turning against their ‘neighbours’ based on an ingroup-outgroup dynamic. We think that people who commit atrocities in these circumstances are the exception, but the exception is the person who refuses, and in fact, saves people who have been demonised.

I think the one thing we can take from Hannah Arendt’s insight - remember she witnessed this firsthand, and was one of the ones who helped – is that evil happens when people do nothing.

 

There is relevance here to the current situation in Australia where we’ve had people in offshore detention for up to 8 years. This is inhumane, to say the least, and most of us do absolutely nothing.

The dialectic between science and philosophy

 This is a consequence of a question I answered on Quora. It was upvoted by the person who requested it, which is very rare for me (might have happened once before).

Naturally, I invoke one of my favourite metaphors. Contrary to what some scientists claim (Stephen Hawking comes to mind) philosophy is not dead, and in fact science and philosophy have a healthy relationship.

 

The original question was:

 

I see a distinction between "Philosophy" which attempts to describe and explain the world and "Science" in which theories can be proven or disproven. Under this paradigm, what areas claiming to be Science are actually Philosophy? (Requested by Michael Wayne Box.)

 

There are, in fact, 98 answers to this question, many by academics, which makes my answer seem pretty pedestrian. But that’s okay; from what I've read, I took a different approach. I’m interested in how science and philosophy interact (particularly epistemology) rather than how they are distinct, though I address that as well. My answer below:

 

 

Science as we know it today is effectively a product of Western philosophy going all the way back to Plato’s Academy and even earlier. In fact, science was better known as ‘natural philosophy’ for much of that time.

 

Science and epistemology have a close relationship, and I would argue that it is a dialectical relationship. John Wheeler provides a metaphor that I find particularly appealing:

 

We live on an island of knowledge in a sea of ignorance. As the island of knowledge expands, so does the shoreline of our ignorance.

 

I came up with this metaphor myself, independently of Wheeler, and I always saw the island of knowledge as science and the shoreline as philosophy. Seen in this light, I contend that science and philosophy have a dialectical relationship. But there is another way of looking at it, and I’d like to quote Russell:

 

Thus, to sum up our discussion of the value of philosophy; Philosophy is to be studied, not for the sake of any definite answers to its question, since no definite answers can, as a rule, be known to be true, but rather for the sake of the questions themselves; because these questions enlarge our conception of what is possible, enrich our intellectual imagination and diminish the dogmatic assurance which closes the mind against speculation; but above all because, through the greatness of the universe, which philosophy contemplates, the mind is also rendered great, and becomes capable of that union with the universe which constitutes its highest good.’ (My emphasis)

 

And this, I believe, touches on the main distinction between science and philosophy: that philosophy asks questions for which we currently don’t have answers and science actually provides answers, even if they’re provisional. And that’s why I argue there is a dialectic, because, as soon as science gives us an answer, it also gives us new questions.

 

To provide examples: Do we live in a multiverse? Will AI become sentient? Was there something before the Big Bang? What is dark matter? How did DNA evolve? All these questions are on the shoreline of our island of knowledge. They are at the boundary between philosophy and science. But that boundary changes as per Wheeler’s metaphor.

 

 

 

That’s what I wrote on Quora, but I can’t leave this subject without talking about the role of mathematics. Curiously, mathematics is also linked to philosophy via Plato and his predecessors. So I will add a comment I wrote on someone else’s post. It might be hubris on my part, but 20^th Century’s preoccupation with language defining our epistemology seems to miss the point, because the Universe speaks to us in its own language, which is mathematical. 

 

With that in mind, this is what I wrote in response to a post that concluded, Philosophy will continue as it has for the past two and a half thousand years, but we presently stand over the wreckage of a philosophical tradition four centuries in the making.

 

 

 

I believe philosophy is alive and well; it’s just had a change of clothes. You can’t divorce philosophy from science, with which it has a dialectical relationship. And that’s just concerning epistemology and ontology.

 

The mistake made in the early 20th Century was to believe mathematics is an artefact created by logic, when in fact, logic is what we use to access mathematics. Godel’s theorem and Turing’s resolution to the halting problem, demonstrate that there will always exist mathematical ‘truths’ that we can’t prove (even if we prove them there will be others). In effect, Godel ‘proved’ (ironically) that there is a difference between ‘truth’ and ‘proof’ in mathematics.

 

But there’s more: Einstein and his cohorts involved in the scientific revolution that took up the whole 20th Century, showed that mathematics lies at the heart of nature. To quote Richard Feynman:

 

Physicists cannot make a conversation in any other language. If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in. She offers her information only in one form.

 

So we have an epistemological link between the natural world and mathematics. It’s not human language that defines what we call reality, but mathematics. From what I read on this subject, it seems philosophers in general still haven’t caught up with this truth.

Implications of the Mary’s Room thought experiment on AI

 This is a question I answered on Quora, mainly because I wanted to emphasise a point that no one discussed. 

This is a very good YouTube video that explains this thought experiment, its ramifications for consciousness and artificial intelligence, and its relevance to the limits of what we can know. I’m posting it here, because it provides a better description than I can, especially if you’re not familiar with it. It’s probably worth watching before you read the rest of this post (only 5 mins).

All the answers I saw on Quora, say it doesn’t prove anything because it’s a thought experiment, but even if it doesn’t ‘prove’ something, it emphasises an important point, which no one discusses, including the narrator in the video: colour is purely a psychological phenomenon. Colour can only exist in some creature’s mind, and, in fact, different species can see different colours that other species can’t see. You don’t need a thought experiment for this; it’s been demonstrated with animal behaviour experiments. Erwin Schrodinger in his lectures, Mind and Matter (compiled into his book, What is Life?), made the point that you can combine different frequencies of light (mix colours, in effect) to give the sensation of a colour that can also be created with one frequency. He points out that this does not happen with sound, otherwise we would not be able to listen to a symphony.

 

The point is that there are experiences in our minds that we can’t share with anyone else and that includes all conscious experiences (a point made in the video). So you could have an AI that can distinguish colours based on measuring the wavelength of reflected light, but it would never experience colours as we do. I believe this is the essence of the Mary's room thought experiment. If you replaced Mary with a computer that held all the same information about colour and how human brains work, it would never have an experience of colour, even if it could measure it.

 

I think the thought experiment demonstrates the difference between conscious experience and AI. I think the boundary will become harder to distinguish, which I explore in my own fiction, but I believe AI will always be a simulation – it won’t experience consciousness as we do.

Grayling railing against God (I couldn’t help myself)

 I’ve just read A.C. Grayling’s book, The God Argument; The Case against Religion and for Humanism (his emphasis). It’s really a polemic against all deistic religions, even though he claims it’s not a polemic, while acknowledging it probably comes across as one. 

His basic argument, which he iterates in many different ways, is that any belief in God or Gods is irrational, starting with the gods of Norse and Greek mythology and including the Biblical God. It’s a sound argument, because, depending on your culture, you tend to treat one variant as fiction and the other as having personal and spiritual significance. Grayling doesn’t address it in this way: instead, arguing that a belief in God is no different to a belief in Santa Claus or the Tooth Fairy; one you grow out of and the other you don’t. The inference is that you are immature or unintelligent or, at best, delusional.

 

I’ve said before that all the Gods I know about have cultural ties and that includes the Abrahamic one. But comparing them to Santa Claus or the Tooth Fairy is like comparing them to fictional characters like Superman and Tarzan, or Luke Skywalker. So, what’s the difference? The difference is in the potency that you give them. A God or Goddess is something internal that only has meaning for you. I’ll return to this idea throughout, because I think that God has no meaning outside someone’s mind. 

 

I rejected the biblical God in my teens, after a childhood spent immersed in its teachings. But the decision was more an emotional one than an analytical one. Grayling acknowledges, by the way, that religious belief is emotional, which, for him, is just another reason to dismiss it. I rejected God because I grew to really, really dislike Him. He was the worst type of tyrant: he ruled by fear and terror; he practised genocide on a global scale (the Noahic flood); he sent his ‘children’ to everlasting torment for disobedience; he tortured Job to win a bet with the Devil. Oh, almost forgot: he was going to get Abraham to kill his son, Isaac, to test his loyalty.

 

I once commented that the question: Does God exist? is the wrong question. The real question, which enters the consciousness of any rational person is: What’s the point? Is there a higher purpose to our existence? This is what religions have attempted to address, and in consequence, some have invoked deities.

 

Grayling, in a philosophical sleight of hand, categorises some Eastern philosophies, like

Buddhism and Confucianism as not being religions, because they don’t invoke gods. I think it’s fair to categorise Confucius as a philosopher in the same mould as teachers like Plato and Aristotle. But, like Jesus, both Buddha and Confucius had disciples, and they were all iconoclasts, challenging the social mores of their day, which they believed to be unfair and iniquitous. In fact, I would put Jesus in the same category as Buddha, Confucius and Socrates, who were all persecuted for challenging the status quo.

 

But Grayling conveniently overlooks that the Chinese, who adopted both Buddhism and Confucianism, culturally worshiped their ancestors, which is surely a religious practice. Perhaps Grayling doesn’t know many Chinese, whereas I have lived with Chinese individuals, and they definitely have deities as part of their traditional culture.

 

I’ve argued previously that science is neutral on the existence of God. In other words, science does not rule out a ‘creator’, yet there is obvious conflict between science and religious texts. Science is an epistemology and religion is not – they don’t compare. Some people argue that religion explains what science cannot, but that’s an argument from ignorance. There will always be things we don’t know – I’ve written extensively on that point – but no religious text can provide an explanation to a question that contemporary science can’t answer.

 

I think the notion of an omniscient God has problems with logic. Clifford A Pickover wrote a very thought-provoking book, The Paradox of GOD and the Science of Omniscience. To give examples: even God doesn’t know the last digit of pi, because it doesn’t exist; and God can’t make a prime number non-prime. Some people argue that God created logic and I argue that God is restrained by logic the same as us. The Universe obeys logic not because God created the logic but because logic transcends the Universe.

 

When I say that science does not rule out ‘God’, I mean it doesn’t rule out a ‘purpose’ that may be beyond our kin. We really don’t know. That doesn’t make me agnostic, as I don’t believe in an anthropomorphic creator, but neither does it mean that people who do, are axiomatically irrational, as Grayling contends. I’ve argued before that, in fact, it’s a non sequitur to believe the God you find inside your mind is the creator of the entire universe.

 

The major problem I have with polemical texts against religion is that, whether intentional or not, they promote intolerance. Back in 2010, I quoted Grayling where he seemed to be promoting religious tolerance.

 

And people who didn't have a religious commitment wouldn't mind if other people did privately and they wouldn't attack or criticise them. 

So there was an unwritten agreement that the matter was going to be left quiet. So in a future where the religious organisations and religious individuals had returned to something much more private, much more inward looking, we might have that kind of public domain where people were able to rub along with one another with much less friction than we're seeing at the moment.

 

I believe this is fundamentally the society I live in (in Australia) where, for the most part, people don’t care what you believe, and where religion is not part of our political landscape. In fact, despite having religion as part of my education, I was brought up with the tacit understanding that religious belief was personal and therefore only shared with others under the mutual understanding that it was confidential and deeply private. A secular society is not an atheistic society; it’s a tolerant society or it doesn’t work. 

 

I know people with completely different religious beliefs to me, best friends, in fact. What’s more, in our current society, I’d say political beliefs are far more divisive than religious beliefs. It puts a lie to the argument, proposed by Grayling and other militant atheists, that if we eliminated religion, ‘at its root’, then we would overcome the world’s conflicts. It’s not only simplistic, but naive, even dangerous. Religion does contribute to conflicts but only when it is politicised, which is what we witness in places where religion demarcates territorial disputes or differences in status. Religion is just one marker of ingroup-outgroup discrimination, with race, language and wealth being more likely contenders.

 

Grayling is contemptuous of people who adapt their religious beliefs to their circumstances, arguing that they ‘cherry-pick’ and are ‘hypocrites’. Well, I readily admit that I cherry-pick all the time - just read my blog - but I don’t see that as hypocrisy.

 

Don Cupitt provides a different perspective, which is the opposite point of view:

 

The only ideas, thoughts, convictions that stay with you and give you real support are ones you have formulated yourself and tested out in your own life… In effect, the only religion that can save you is one you have made up for yourself and tested out for yourself: in short, a heresy.

 

Grayling addresses the teleological arguments and the ontological argument and the cosmological argument, all in some detail, which I won’t go into. Paul Davies spent considerable time on them as well in his book, The Mind of God. 

 

But there is one argument that Grayling addressed which I found interesting, and that was Plantinga’s version of the ontological argument based on modal logic. I’ve come across this before, which is based on the premise that if something necessarily exists in a possible world then it must exist in all possible worlds (my emphasis). The problem is with the premise that God must necessarily exist in a possible world. I’ve always thought that this argument is somewhat circular, because it seems to assume that God necessarily exists, which is what it’s trying to prove, via logic alone. Grayling goes into it in some detail and claims that Plantinga eventually gave it up, falling back on an even less credible argument that we know that God exists in the same way we know that the past exists. I may have oversimplified it, but that’s the analogy that Grayling used.

 

The teleological argument comes from Aristotle, as Grayling expounds, because he argued that everything manmade has a ‘final cause’, which is the cause that prompted someone to make it, and you could apply this to the whole universe. I have my own response to this. If humans are the ‘final cause’ of God’s ‘creation’, then, without humans, God has no reason to exist. And this leads me to argue a reverse logic that God is dependent on humans rather than the other way round.

 

This is related to the fine-tuned argument that the Universe is ‘just right’ for complex life to emerge and leads to the anthropic principle. Grayling doesn’t mention the anthropic principle, probably because it tacitly allows teleology back into the picture. Grayling makes an analogy by saying that his antecedents only existed so he could exist, which is a good argument. But the point I like to make is that without conscious entities, the Universe may as well not exist. And we are special in as much as we have the unique ability to comprehend the Universe, as Einstein famously pointed out. Or, as Paul Davies said, ‘we can unravel the plot’. The alternative is what Davies calls the ‘absurd universe’, which appears to be the one Grayling plumps for: we give it a meaning because we are predisposed to providing meanings, but there is no reason to think one should exist. 

 

But this goes to the heart of the debate for me. The whole reason we have religion of any type is because humans wondered if there was something beyond the mortal realm. No one can answer that, but it’s why we created gods in all their manifestations. So gods become a part of a collective consciousness, which is why they can seem real to us. In this context, God is a projection that we laden with all our prejudices and hopes beyond death. One cannot dissociate any notion of God from the human psyche, as Ludwig Feuerbach pointed out in the 19^th Century. God is always in human image, not the other way round. Grayling doesn’t discuss this at all, but I fail to see how one can address God independently of a human context. In effect, we get the God we deserve. And by God, I mean the ideal we imagine we should aspire to. This is why we deify mortal humans like Jesus and Buddha, because they represent an ideal that they could only achieve beyond death.

 

The second half of his book talks about humanism. He spends a chapter on the importance and interdependence of authenticity and truth, and another on human rights. They remind me of my 3 rules for humans. He spends an entire chapter on the ethics of sexual conduct and how it’s been perverted by civilised societies. The book is worth acquiring for that alone.

 

Anyone who reads my blog, knows that I think God is subjective, not objective. Anyone who is a believer, will tell you that God came to them, meaning that God only exists in their mind, not out there. I have no issue with this idea of God; but it’s not what religions tell you. Anyone who has a religious experience is an iconoclast, including Jesus and Buddha. I think the idea that God evolves as a product of our consciousness is far more logical than the idea that He (why he?) created us in his image, as potential companions.

 

I make a distinction between non-theists and atheists. In Australia, there are a lot of non-theists, meaning they don’t care what you believe. Going by this tome (2013), Grayling is ‘anti-theist’, though he claims it’s not a religious belief; it’s the opposite of belief. However, his polemic indicates that he cares about whether someone believes in God or not and, like Dawkins, Harris and others, he proselytises atheism. This is not a non-theistic attitude. Anti-theism may not be a religion, but it’s anti-religious in its rhetoric.

 

I will leave the last word to Einstein, who talks about religion with no mention of God.

 

The most beautiful and deepest experience a man can have is the sense of the mysterious. It is the underlying principle of religion as well as of all serious endeavour in art and science. He who never had this experience seems to me, if not dead, then at least blind. To sense that behind anything that can be experienced there is a something that our minds cannot grasp, whose beauty and sublimely reaches us only indirectly: this is religiousness. In this sense I am religious. To me it suffices to wonder at these secrets and to attempt humbly to grasp with my mind a mere image of the lofty structure of all there is.

What’s the most fundamental value?

 This is a Question of the Month in Philosophy Now (Issue 143, April/May 2021). I wrote it very quickly, almost on impulse in less than ½ hr, but I spent a lot of time polishing it.

The word ‘fundamental’ is key here: it infers the cornerstone or foundation upon which all other values are built. Carlo Rovelli, who is better known as a physicist than a philosopher, said in an online video that “we are not entities, we are relations”. And I believe this aphorism goes to the heart of what it means to be human. From our earliest cognitive moments to the very end of our days, the quality of our lives is largely dependent on our relationships with others. And, in that context, I would contend that the most important and fundamental value is trust. Without trust, honesty does not have a foothold, and arguably honesty is the glue in any relationship, be it familial, contractual or even between governments and the general public. 

 

Psychologists will tell you that fear and trust cannot co-exist. If someone, either as a child, or a spouse, is caught in a relationship governed by fear, yet completely dependent, the consequence will inevitably result in an inability to find intimacy outside that relationship, because trust will be corroded if not destroyed. 

 

Societies can’t function without trust: traffic would be chaos; projects wouldn’t be executed collaboratively. We all undertake financial transactions every day and there is a strong element of trust involved in all of these that most of us take for granted. Cynics will argue that trust allows others to take advantage of you, which means trust only works if it is reciprocated. If enough people take advantage of those who trust, then it would evaporate and everyone would suddenly dissemble and obfuscate. Relationships would be restricted to one’s closest family and wider interactions would be fraught with hidden agendas, even paranoia. But this is exactly what happens when governments mandate their citizenry to ‘out’ people who don’t toe the party line. 

 

Everything that we value in our relationships and friendships, be it love, integrity, honesty, loyalty or respect, is forfeit without trust. As Carlo Rovelli intimated in his aphoristic declaration, it is through relationships that we are defined by others and how we define ourselves. It is through these relationships that we find love, happiness, security and a sense of belonging. We ultimately judge our lives by the relationships we form over time, both in our professional lives and our social lives. Without trust, they simply don’t exist, except as fake.

                                                --------------------------

I once wrote on this topic before, in 2008. I deliberately avoided reading that post while I wrote this one. To be honest, I’m glad I did as it’s a much better post. However, this is a response to a specific question with a limit of 400 words. Choosing the answer was the easy part – it took seconds – arguing a case was more organic. I’ll add an addendum if it’s published.

Interestingly, 'trust' crops up in my fiction more than once. In the last story I wrote, it took centre stage.

Philosophy of mathematics

 I’ve been watching a number of YouTube videos on this topic, although some of them are just podcasts with a fixed-image screen – usually a blackboard of equations. I’ll provide links to the ones I feel most relevant. I’ve discussed this topic before, but these videos have made me reassess and therefore re-analyse different perspectives. My personal prejudice is mathematical Platonism, so while I’ll discuss other philosophical positions, I won’t make any claim to neutrality.

What I’ve found is that you can divide all the various preferred views into 3 broad categories. Mathematics as abstract ‘objects’, which is effectively Platonism; mathematics as a human construct; and mathematics as a descriptive representation of the physical world. These categories remind one of Penrose’s 3 worlds, which I’ve discussed in detail elsewhere. None of the talks I viewed even mention Penrose, so henceforth, neither will I. I contend that all the various non-Platonic ‘schools’, like formalism, constructivism, logicism, nominalism, Aristotlean realism (not an exhaustive list) fall into either of these 2 camps (mental or physical attribution) or possibly a combination of both. 

 

So where to start? Why not start with numbers, as at least a couple of the videos did. We all learn numbers as children, usually by counting objects. And we quickly learned that it’s a concept independent of the objects being counted. In fact, many of us learned the concept by counting on our fingers, which is probably why base 10 arithmetic is so universal. So, in this most simple of examples, we already have a combination of the mental and the physical. I once made the comment on a previous post that humans invented numbers but we didn’t invent the relationships between them. More significantly, we didn’t specify which numbers are prime and which are non-prime – it’s a property that emerges independently of our counting or even what base arithmetic we use. I highlight primes, in particular, because they are called ‘the atoms of mathematics’, and we can even prove that they go to infinity.

 

But having said that, do numbers exist independently of the Universe? (As someone in one of the videos asked.) Ian Stewart was the first person I came across who defined ‘number’ as a concept, which infers they are mental constructs. But, as pointed out in the same video, we have numbers like pi which we can calculate but which are effectively uncountable. Even the natural numbers themselves are infinite and I believe this is the salient feature of mathematics. Anything that’s infinite transcends the Universe, almost by definition. So there will always be aspects of mathematics that will be unknowable, yet, we can ‘prove’ they exist, therefore they must exist outside of space and time. In a nutshell, that’s my best argument for mathematical Platonism.

 

But the infinite nature of mathematics means that even computers can’t deal with a completely accurate version of pi – they can only work with an approximation (as pointed out in the same video). This has led some mathematicians to argue that only computable numbers can be considered part of mathematics. Sydney based mathematician, Norman Wildberger, provides the best arguments I’ve come across for this rather unorthodox view. He claims that the Real numbers don’t exist, and is effectively a crusader for a new mathematical foundation that he believes will reinvent the entire field.

Probably the best talk I heard was a podcast from The Philosopher’s Zone, which is a regular programme on ABC Radio National, where presenter, Alan Saunders, interviewed James Franklin, Professor in the School of Mathematics and Statistics at UNSW (University of New South Wales). I would contend there is a certitude in mathematics we don't find in other fields of human endeavour. Freeman Dyson once argued that a mathematical truth is for all time – it doesn’t get overturned by subsequent discoveries.

 

And one can’t talk about mathematical ‘truth’ without talking about Godel’s Incompleteness Theorem. Godel created a self-referencing system of logic, whereby he created the mathematical equivalent of the ‘liar paradox’ – ‘this statement is false’. He effectively demonstrated that within any ‘formal’ system of mathematics you can’t prove ‘consistency’. This video by Mark Colyvan (Professor of Philosophy and Director of the Sydney Centre for the Foundations of Science), explains it better than I can. I’m not a logician, so I’m not going to expound on something I don’t fully understand, but the message I take from Godel is that he categorically showed there is a fundamental difference between ‘truth’ and ‘proof’ in mathematics. Basically, in any axiom-based mathematical system (that is consistent), there exist mathematical ‘truths’ that can’t be proved. It’s the word axiom that is the key, because, in principle, if one extends the axioms one can possibly find a proof.

 

Extending axioms extends mathematics, which is what we’ve done historically since the Ancient Greeks. I referenced Norman Wildberger earlier, and what I believe he’s attempting with his ‘crusade’, is to limit the axioms we’ve adopted, although he doesn’t specifically say that.

 

Someone on Quora recently claimed that we can have ‘contradictory axioms’, and gave Euclidian and subsequent geometries as an example. However, I would argue that non-Euclidean (curved) geometries require new axioms, wherein Euclidean (flat) geometry becomes a special case. As I said earlier, I don’t believe new discoveries prove previous discoveries untrue; they just augment them.

 

But the very employment of axioms, begs a question that no one I listened to addressed: didn’t we humans invent the axioms? And if the axioms are the basis of all the mathematics we know, doesn’t that mean we invented mathematics?

 

Let’s look at some examples. As hard as it is to believe, there was a time when mathematicians were sceptical about negative numbers in the same way that many people today are sceptical about imaginary numbers (i = √-1). If you go back to the days of Plato and his Academy, geometry was held in higher regard than arithmetic, because geometry could demonstrate the ‘existence’, if not the value, of incalculable numbers like π and √2. But negative numbers had no meaning in geometry: what is a negative area or a negative volume?

 

But mathematical ‘inventions’ like negative numbers and imaginary numbers allowed people to solve problems that were hitherto unsolvable, which was the impetus for their conceptual emergence. In both of these cases and the example of non-Euclidean geometry, whole new fields of mathematics opened up for further exploration. But, also, in these specific examples, we were adding to what we already knew. I would contend that the axioms themselves are part of the exploration. If one sees the Platonic world of mathematics as a landscape that only sufficiently intelligent entities can navigate, then axioms are an intrinsic part of the landscape and not human projections.

 

And, in a roundabout way, this brings me back to my introduction concerning the numbers that we discovered as children, whereby we saw a connection between an abstract concept and the physical world. James Franklin, whom I referenced earlier, gave the example of how we measure an area in our backyard to determine if we can fit a shed into the space, thereby arguing the case that mathematics at a fundamental level, and as it is practiced, is dependent on physical parameters. However, what that demonstrates to me is that mathematics determines the limits of what’s physically possible and not the other way round. And this is true whether you’re talking about the origins of the Universe, the life-giving activity of the Sun or the quantum mechanical substrate that underlies our entire existence.

Footnote: Daniel Sutherland (Professor of Philosophy at the University of Illinois, Chicago) adopts the broad category approach that I did, only in more detail. He also points out the 'certainty' of mathematical knowledge that I referenced in the main text. Curiously, he argues that the philosophy of mathematics has influenced the whole of Western philosophy, historically.