I’ve just read a recently published book, The Universe Speaks in Numbers; How Modern Maths Reveals Nature’s Deepest Secrets by Graham Farmelo. The title says it all really, but the content takes you places you’ve probably never been, intellectually. This book, possibly more than any other book I’ve read, gives you access to the minds of the superstars in theoretical physics, both past and present, but mostly present. It’s like listening to a group of virtuoso musicians and becoming aware of how inadequate you are in the presence of such talent.
It’s written in an historical context, from Newton through Maxwell to Einstein, but rather than lingering in the so-called golden age of physics, it focuses on the last 50 years.
Farmelo covers what he calls ‘the long divorce’, a term coined by Freeman Dyson, covering much of the latter 20th Century, when pure mathematicians and theoretical physicists, not only avoided each other, but were not very accommodating of each other’s contributions.
That all changed very late in the 20th Century, and since then there has been a synergy that has benefited both disciplines. The problem is that theoretical physics has leapfrogged ahead of experimental physics, and, consequently, physicists at the leading edge of high energy physics have become more dependent on mathematical solutions to provide veracity to their ideas. ‘High energy’ means highly confined dimensions, so we are talking about the smallest constituents of matter we currently know or can imagine.
I’ve made the point in previous posts that the ultimate arbiter of truth is evidence, but Farmelo’s book has made me re-evaluate that position. Instead of testing theories with experiments that can’t be performed, theoretical physicists simply adhere to 2 founding principles: quantum mechanics and special relativity. Because those 2 specific theories have been experimentally tested in extremis, they simply ensure that any new theory, no matter at what level, or using whatever mathematical tools they have, must meet both criteria.
The book necessarily covers string theory, from its inception to its current status. But what I found most fascinating was all the discoveries that seemed to have occurred on the side so-to-speak, where mathematics appears to continually underlie the fundamentals of nature in unexpected ways.
I admit to being a sceptic of string theory, due to its 6 additional dimensions that can’t be observed and its plethora of 10500 universes. But I must also admit that the people exploring this mathematical world leave me stranded when it comes to intellectual wizardry.
Farmelo repeatedly refers to 2 talks given by Albert Einstein and Paul Dirac, respectively, where they effectively gave a call-to-arms, arguing that mathematics is the key to new theories in physics, with experimental physics providing confirmation rather than having the leading role. Whether by design or accident, this is how physics has evolved in the last 5 decades. As Farmelo expounds, not everyone has been happy with this development, yet there have been successes.
To give one example, mathematical devices called twistors (developed by Roger Penrose in the 1960s) have led to providing accurate predictions in the amplitudes of scattering gluons, which are the mediating particles for quarks in atomic nuclei. This short description belies the convoluted story, involving many theorists in the UK and the US, and the many unexpected discoveries made along the way; including a connection with a mathematical object discovered by Hermann Grassmann in 1844, called eponymously a Grassmannian. It led to another mathematical object called an 'amplituhedron'. One of the co-discoverers, Nima Arkani-Hamed (an American born Iranian, at the Princeton Institute for Advanced Study) said:
This is a concrete example of a way in which the physics we normally associate with space-time and quantum mechanics arises from something more basic.
The ‘something more basic’ is only known mathematically, as opposed to physically. I found this a most compelling tale and a history lesson in how mathematics appears to be intrinsically linked to the minutia of atomic physics.
In the same context, Arkani-Hamed says that ‘the mathematics of whole numbers in scattering-amplitude theory chimes… with the ancient Greeks' dream: to connect all nature with whole numbers.’
There is an assumption by non-physicists that the role of mathematics in understanding nature is a consequence of the fact that we need to measure everything. A common criticism is that people who emphasise the role of mathematics in their theories have ‘mistaken the map for the terrain’.
Einstein was probably the first to use mathematics alone to sculpture a theory independently of observation and experimentation, when he developed his masterpiece, the general theory of relativity. It was his mathematical prediction that gravity would bend light that clinched his theory when few people believed that relativity reflected reality.
In reference to the abovementioned metaphor about the ‘map and terrain’, there is an axiomatic inference that the map is derived from ‘surveying’ the terrain. However, it’s becoming increasingly apparent that the map is discovered before the terrain is even 'explored', which turns the metaphor on its head. It’s not a metaphor I would choose to use, but if you insist, you might have to consider the possibility that the map pre-exists the terrain.
In reference to the title, I’ll retell a joke by mathematical physicist, Robbert Dijkgraaf, from the Princeton Institute for Advanced Study:
What is the difference between a physicist and a mathematician?
A physicist studies the laws that God chose for nature to obey.
A mathematician studies the laws that God has to obey.
02 August 2019
24 July 2019
Sometimes being paranoid is healthy
Not that long ago I wrote a post about how the technology of surveillance and its ever-increasing role in policing was overtaking fiction. As a sometime science fiction writer, it comes with the territory to explore future societies. In my fiction, I really don’t attempt to forecast the future, I simply use a fictional social landscape to explore ideas and, in particular, relationships with advanced technology. There is an element of fantasy in my stories as well.
All governments know the importance of controlling information, called ‘controlling the narrative’, which is why an independent media is essential to a democracy, and it’s also why the most totalitarian governments find ways to imprison journalists who challenge the ‘party narrative’. In the current political climate throughout the Western world, opinions have become polarised on almost every issue, and, as someone recently pointed out, media outlets have become default political mouthpieces. Followers on both sides of politics now want to silence or muzzle media sources that disagree with their particular political point of view.
Very recently, in Australia, following a surprising election win by an incumbent conservative government, media outlets were raided (both public and commercial sources) including an award-winning female journalist’s home. This rang alarm bells from all sides of media. The timing was significant because the 2 different news stories went to air more than a year earlier, yet the raids occurred within a week of winning an election. The Government claims the raids were independent of them; so aren’t they lucky the raids didn’t occur the week before the election instead of the week following.
In the case of the ABC (Australian Broadcasting Corporation), Federal Police officers sat in a room and went through emails and edited them, apparently. Effectively changing records to suit their purpose. If this happens in a democracy, what happens in totalitarian regimes. (Well, journalists are put in jail or assassinated in some cases, like Russia).
Now imagine a future society where everyone is tracked by facial recognition and everyone has a social ‘rating’ which determines what buildings they can enter, what job they can apply for and even what transport they can board, be it a train or a plane. Did I say future? This is already happening in the most populous nation in the world. It’s well known that this same nation controls very exactly what information people can access.
It was less than 20 years ago, when there was a ‘Y2K’ scare, predicting that important utilities and infrastructure would fail when computer clocks ticked over to the year 2000, because their clocks didn’t go beyond that date. As it turned out, the scare was ill-founded, but it highlighted how dependent we are on the internet infrastructure, and how vulnerable it could be in a future transcontinental conflict.
Most people are blissfully unaware of how our computing systems are based on 2 formats that go back decades – Microsoft and Apple – so it would be virtually impossible to introduce a new system from scratch. But what if the internet infrastructure became just as dependent on one system, and as their monopoly grew over the entire world, how hard it would be to ‘disconnect’.
Now, also imagine if the infrastructure that everyone depended on was run by one of the most totalitarian regimes in the world, who had a paranoid obsession with the control of information.
All governments know the importance of controlling information, called ‘controlling the narrative’, which is why an independent media is essential to a democracy, and it’s also why the most totalitarian governments find ways to imprison journalists who challenge the ‘party narrative’. In the current political climate throughout the Western world, opinions have become polarised on almost every issue, and, as someone recently pointed out, media outlets have become default political mouthpieces. Followers on both sides of politics now want to silence or muzzle media sources that disagree with their particular political point of view.
Very recently, in Australia, following a surprising election win by an incumbent conservative government, media outlets were raided (both public and commercial sources) including an award-winning female journalist’s home. This rang alarm bells from all sides of media. The timing was significant because the 2 different news stories went to air more than a year earlier, yet the raids occurred within a week of winning an election. The Government claims the raids were independent of them; so aren’t they lucky the raids didn’t occur the week before the election instead of the week following.
In the case of the ABC (Australian Broadcasting Corporation), Federal Police officers sat in a room and went through emails and edited them, apparently. Effectively changing records to suit their purpose. If this happens in a democracy, what happens in totalitarian regimes. (Well, journalists are put in jail or assassinated in some cases, like Russia).
Now imagine a future society where everyone is tracked by facial recognition and everyone has a social ‘rating’ which determines what buildings they can enter, what job they can apply for and even what transport they can board, be it a train or a plane. Did I say future? This is already happening in the most populous nation in the world. It’s well known that this same nation controls very exactly what information people can access.
It was less than 20 years ago, when there was a ‘Y2K’ scare, predicting that important utilities and infrastructure would fail when computer clocks ticked over to the year 2000, because their clocks didn’t go beyond that date. As it turned out, the scare was ill-founded, but it highlighted how dependent we are on the internet infrastructure, and how vulnerable it could be in a future transcontinental conflict.
Most people are blissfully unaware of how our computing systems are based on 2 formats that go back decades – Microsoft and Apple – so it would be virtually impossible to introduce a new system from scratch. But what if the internet infrastructure became just as dependent on one system, and as their monopoly grew over the entire world, how hard it would be to ‘disconnect’.
Now, also imagine if the infrastructure that everyone depended on was run by one of the most totalitarian regimes in the world, who had a paranoid obsession with the control of information.
21 July 2019
Religion and politics in secular society
This is a letter I wrote to The Weekend Australian, Christmas 2017, so 18 months ago. There was a side-debate at the time, during the same sex marriage debate about ‘religious freedom’. It seemed to me that ‘religious freedom’ was ‘code’ for freedom to discriminate against gays and lesbians, and it’s not over. From what I’ve read and seen, the only arguments against same sex marriage came from conservative religious figures (some being politicians) even if they claimed it was about ‘the sanctity of marriage’.
The letter below alludes to that debate, even though the topic is much broader. It’s really about a perceived conflict between secularism and Christianity in Western societies, including Australia. There is a recurring argument that our Christian heritage provides the moral fabric of our society with the inference that, without it, we’d lose our moral compass. If that was true, and we really followed Christ’s calling, we wouldn’t treat refugees the way we do. In fact, our 2 most conservative Christian leaders, in recent times, have been the most ruthless advocates for persecuting refugees, and of fomenting xenophobic sentiment in the electorate.
The names referenced in the letter below, are journalists or commentators. The Australian is a Murdoch publication, so it has conservative political leanings.
Both Paul Kelly and John Carroll in separate articles (Weekend Australian, Enquirer, 23-24 Dec.2017) seem concerned that the modern secular world that dominates Western societies, and therefore Australia, has forgotten, even ‘turned its back’ on our Christian heritage. I’m officially retiring age, so I grew up in post-war Australia when going to Sunday School and scripture classes (in public schools) was still considered part of a child’s education (neither of my parents were religious; they just thought it was the cultural norm). Strangely, I don’t lament the loss, for want of a better word, the Church’s role in political and secular life, epitomised by the divide between Catholics and Protestants that dominated even the small country town where I grew up.
I found the greater part of Paul Kelly’s lengthy editorial a stimulating read, even when I might proffer alternative views, but his commentary on High Court judge Dyson Heydon’s concerns about the future of Christianity in this country, I found alarmist to say the least: “The question for the West is how it retains its civilisational heritage if it abandons beliefs in its Christian ethic or, indeed, if its political culture begins to assault that ethic.” Without referencing them specifically, he’s obviously referring to the passing of the recent same-sex-marriage bill in Federal Parliament and the euthanasia bill in the Victorian State Parliament. Both of these have provoked ‘concerns’ from the Catholic church, in particular, who are effectively under siege for the sins they committed in the previous generation.
Personally, I think it’s a landmark moment that gays and lesbians now have the same rights as heterosexual couples. A law that has symbolic and pragmatic importance for the people it affects, and absolutely no effect on the people who oppose it. No one is being forced to have same-sex marriage – it’s a choice. Kelly and his fellow detractors will talk about religious freedom, but it’s only an issue for the people who, for whatever reason, think that homosexuals and lesbians should stay in the closet, or at least, stay out of our churches. If it comes to a choice – and it shouldn’t – between gay and lesbian rights and religious freedom, then it’s a no brainer for most Australians, including the ones like myself, who are heterosexual.
To give credit to The Australian, on the same page as John Carroll’s very lengthy piece on Christmas and the declining relevance of Jesus’ story to most Australians, there is a piece by Helen Dale, who explains, at some length, the pagan roots of Christmas that most Australians are either unaware of or blissfully apathetic about. This created a counter-perspective that was running through my head even while I was reading Carroll’s thesis.
Don Cupitt, a theologian, turned philosopher and author, is a bit of an iconoclast when it comes to religion and Christianity in particular. He’s made the salient point that humanist morality really started with the novel, where moral dilemmas and issues concerning good and evil were resolved without invoking a Deity or scripture. Carroll, to his credit, makes a similar point about the role of literature and popular culture in stimulating our psyche in this regard, without resort to prescriptive Christian ethics. He then goes on to say: “Further, we have now had 150 years of gloomy prediction that the death of God would lead to political anarchy and the moral collapse of the West. That has simply not eventuated.”
For Carroll, the Jesus story is all about imbibing us with meaning, and that is what we are losing. The point is that the Jesus story is mythology and when I was a child, undergoing the religious education I mentioned in my introductory paragraph, I really believed the stories were true, because at that age we believe whatever adults tell us. Like many of my generation, I grew up disillusioned in my mid teens, when I realised the stories were not only mythologised but defied rational analysis. And that is the real reason that Christianity has lost its meaning for most people with a Western education.
In the Review section of the same issue of the Weekend Australian, John Carey reviews a book by Stephen Greenblatt, The Rise and Fall of Adam and Eve, where he charters the literary history of the book of Genesis, from its origins in The Epic of Gilgamesh to Augustine’s seminal re-interpretation as signifying the ‘Fall of Man’. I’m not sure if it’s Carey’s or Greenblatt’s insight, but one of them points out the logical inconsistency in the morality tale: “For if God was all-knowing, why did he forbid Adam and Eve to eat the fruit, knowing they would disobey? Why did he create them at all, since he intended to kill them?”
Both Carroll and Kelly refer to the heritage or legacy that Christian ethics has provided to Western cultures. Well, historically, so-called Christian ethics has created a lot of bigotry, wars, genocide and inquisitional torture. Many contemporary commentators point to the current issues surrounding Islam, claiming that the religion itself is flawed. Well, if Islam is flawed then so is Christianity.
Hugh Mackay, in his book, Right & Wrong; How to Decide for Yourself, warns of the dangers of believing that God is on your side, because then anything can be justified, which is what we’ve witnessed both historically and contemporarily.
Addendum: The same day I posted this, I read an article in the Australian Weekend Magazine (July 20-21, 2019) about Israel Folau and the issue of religious freedom. Folau is a star rugby player with the Australian Rugby Union team and famously posted a piece on Facebook that all homosexuals, adulterers, liars and various other sinners would go to Hell. What created a furor wasn’t so much what he said but that he was sacked from the team (his contract was terminated). I agree with former Australian Human Rights Commissioner, Gillian Triggs, that the Australian Rugby Union went outside their remit.
The incident has brought out the worst on both sides of the debate, and demonstrates what happens when you try to enforce what people are allowed to say in public. Peter Singer is another unexpected supporter of Folau's right to free speech. My attitude is that everyone should be allowed to make complete fools of themselves, whether they be sports stars, TV celebrities, politicians or even the President of the United States.
The letter below alludes to that debate, even though the topic is much broader. It’s really about a perceived conflict between secularism and Christianity in Western societies, including Australia. There is a recurring argument that our Christian heritage provides the moral fabric of our society with the inference that, without it, we’d lose our moral compass. If that was true, and we really followed Christ’s calling, we wouldn’t treat refugees the way we do. In fact, our 2 most conservative Christian leaders, in recent times, have been the most ruthless advocates for persecuting refugees, and of fomenting xenophobic sentiment in the electorate.
The names referenced in the letter below, are journalists or commentators. The Australian is a Murdoch publication, so it has conservative political leanings.
Both Paul Kelly and John Carroll in separate articles (Weekend Australian, Enquirer, 23-24 Dec.2017) seem concerned that the modern secular world that dominates Western societies, and therefore Australia, has forgotten, even ‘turned its back’ on our Christian heritage. I’m officially retiring age, so I grew up in post-war Australia when going to Sunday School and scripture classes (in public schools) was still considered part of a child’s education (neither of my parents were religious; they just thought it was the cultural norm). Strangely, I don’t lament the loss, for want of a better word, the Church’s role in political and secular life, epitomised by the divide between Catholics and Protestants that dominated even the small country town where I grew up.
I found the greater part of Paul Kelly’s lengthy editorial a stimulating read, even when I might proffer alternative views, but his commentary on High Court judge Dyson Heydon’s concerns about the future of Christianity in this country, I found alarmist to say the least: “The question for the West is how it retains its civilisational heritage if it abandons beliefs in its Christian ethic or, indeed, if its political culture begins to assault that ethic.” Without referencing them specifically, he’s obviously referring to the passing of the recent same-sex-marriage bill in Federal Parliament and the euthanasia bill in the Victorian State Parliament. Both of these have provoked ‘concerns’ from the Catholic church, in particular, who are effectively under siege for the sins they committed in the previous generation.
Personally, I think it’s a landmark moment that gays and lesbians now have the same rights as heterosexual couples. A law that has symbolic and pragmatic importance for the people it affects, and absolutely no effect on the people who oppose it. No one is being forced to have same-sex marriage – it’s a choice. Kelly and his fellow detractors will talk about religious freedom, but it’s only an issue for the people who, for whatever reason, think that homosexuals and lesbians should stay in the closet, or at least, stay out of our churches. If it comes to a choice – and it shouldn’t – between gay and lesbian rights and religious freedom, then it’s a no brainer for most Australians, including the ones like myself, who are heterosexual.
To give credit to The Australian, on the same page as John Carroll’s very lengthy piece on Christmas and the declining relevance of Jesus’ story to most Australians, there is a piece by Helen Dale, who explains, at some length, the pagan roots of Christmas that most Australians are either unaware of or blissfully apathetic about. This created a counter-perspective that was running through my head even while I was reading Carroll’s thesis.
Don Cupitt, a theologian, turned philosopher and author, is a bit of an iconoclast when it comes to religion and Christianity in particular. He’s made the salient point that humanist morality really started with the novel, where moral dilemmas and issues concerning good and evil were resolved without invoking a Deity or scripture. Carroll, to his credit, makes a similar point about the role of literature and popular culture in stimulating our psyche in this regard, without resort to prescriptive Christian ethics. He then goes on to say: “Further, we have now had 150 years of gloomy prediction that the death of God would lead to political anarchy and the moral collapse of the West. That has simply not eventuated.”
For Carroll, the Jesus story is all about imbibing us with meaning, and that is what we are losing. The point is that the Jesus story is mythology and when I was a child, undergoing the religious education I mentioned in my introductory paragraph, I really believed the stories were true, because at that age we believe whatever adults tell us. Like many of my generation, I grew up disillusioned in my mid teens, when I realised the stories were not only mythologised but defied rational analysis. And that is the real reason that Christianity has lost its meaning for most people with a Western education.
In the Review section of the same issue of the Weekend Australian, John Carey reviews a book by Stephen Greenblatt, The Rise and Fall of Adam and Eve, where he charters the literary history of the book of Genesis, from its origins in The Epic of Gilgamesh to Augustine’s seminal re-interpretation as signifying the ‘Fall of Man’. I’m not sure if it’s Carey’s or Greenblatt’s insight, but one of them points out the logical inconsistency in the morality tale: “For if God was all-knowing, why did he forbid Adam and Eve to eat the fruit, knowing they would disobey? Why did he create them at all, since he intended to kill them?”
Both Carroll and Kelly refer to the heritage or legacy that Christian ethics has provided to Western cultures. Well, historically, so-called Christian ethics has created a lot of bigotry, wars, genocide and inquisitional torture. Many contemporary commentators point to the current issues surrounding Islam, claiming that the religion itself is flawed. Well, if Islam is flawed then so is Christianity.
Hugh Mackay, in his book, Right & Wrong; How to Decide for Yourself, warns of the dangers of believing that God is on your side, because then anything can be justified, which is what we’ve witnessed both historically and contemporarily.
Addendum: The same day I posted this, I read an article in the Australian Weekend Magazine (July 20-21, 2019) about Israel Folau and the issue of religious freedom. Folau is a star rugby player with the Australian Rugby Union team and famously posted a piece on Facebook that all homosexuals, adulterers, liars and various other sinners would go to Hell. What created a furor wasn’t so much what he said but that he was sacked from the team (his contract was terminated). I agree with former Australian Human Rights Commissioner, Gillian Triggs, that the Australian Rugby Union went outside their remit.
The incident has brought out the worst on both sides of the debate, and demonstrates what happens when you try to enforce what people are allowed to say in public. Peter Singer is another unexpected supporter of Folau's right to free speech. My attitude is that everyone should be allowed to make complete fools of themselves, whether they be sports stars, TV celebrities, politicians or even the President of the United States.
30 June 2019
What does logic reveal about reality?
This is about a loop in our universe (that includes us), and which I’ve long been fascinated by.
To quote from another post I wrote, The introspective cosmos:
We are each an organism with a brain that creates something we call consciousness that allows us to reflect on ourselves, individually. And the Universe created, via an extraordinary convoluted process, the ability to reflect on itself, its origins and its possible meaning.
This insight is also reflected in Eugene Wigner’s 2 miracles: the miracle that the Universe can be comprehended and the miracle that we have the ability to comprehend it to the degree that we do. Or as Einstein so famously said:
The most incomprehensible thing about the Universe is that it’s comprehensible.
As Wigner explicitly stated and Einstein implicitly believed, the medium for that comprehension is mathematics. This loop, that I alluded to in my opening, is also implicit in Roger Penrose’s 3 worlds.
The question in the title was one I found on Quora. Most of the questions that Quora’s algorithms address to me are either too silly, or too specialist and esoteric for my capabilities to respond.
In this case, after reading the other answers, I thought they had largely missed the mark, and perhaps the point. The authors may draw the same conclusion about my answer.
I found that my answer went in a subtly different direction to what I intended, but resulted in a mini-epiphany. There is a limit to what we can know because there will always be a limit to the mathematics we know, which thus far determines what we know of the cosmos.
My answer to What does logic reveal about reality?
Fundamentally, it reveals that there are limits to what we can know.
Epistemology is the ‘theory of knowledge’ (dictionary definition) - effectively, the study of what we can know. Whereas ontology is defined as ‘the nature of being’, which, in effect, is what we call reality.
Since the Enlightenment, it’s become increasingly apparent that it’s our knowledge of mathematics that determines the limits of what we can know, both at the cosmological and the infinitesimal scale. But mathematics itself has epistemological limits according to Godel’s Incompleteness Theorem.
In effect, Godel proved that, in any axiom based mathematical system, there will be mathematical truths that we can’t prove. In practice, this means that there will always be mathematical truths that lie beyond what we currently know. In this context, ‘what we currently know’ is transient. So even though we may, and will, know more in the future, it will never be complete.
The point is that we use logic to reveal these mathematical truths and so the corollary to Godel’s theorem is that there will always be a limit to what that logic can reveal, no matter how much it has revealed already. Basically, we extend our knowledge by extending our axiomatic system. To give an example: by employing the new axiom, √-1 = i, we uncovered a whole new realm of mathematics.
Some centuries later, we then used that particular mathematics (called complex algebra) to describe a newly discovered phenomena called quantum mechanics (QM). In fact, without that knowledge (revealed by pure logic) quantum mechanics would never have been developed into a consistent and highly successful theory. And arguably, QM is ‘the evanescent substrate on which we all exist’ [or reality] to quote Clifford A Pickover.
And this is the loop: QM is the substrate of the Universe, which created humans which discovered an abstract mathematics, which not only describes, but prescribes the rules for QM.
To quote from another post I wrote, The introspective cosmos:
We are each an organism with a brain that creates something we call consciousness that allows us to reflect on ourselves, individually. And the Universe created, via an extraordinary convoluted process, the ability to reflect on itself, its origins and its possible meaning.
This insight is also reflected in Eugene Wigner’s 2 miracles: the miracle that the Universe can be comprehended and the miracle that we have the ability to comprehend it to the degree that we do. Or as Einstein so famously said:
The most incomprehensible thing about the Universe is that it’s comprehensible.
As Wigner explicitly stated and Einstein implicitly believed, the medium for that comprehension is mathematics. This loop, that I alluded to in my opening, is also implicit in Roger Penrose’s 3 worlds.
The question in the title was one I found on Quora. Most of the questions that Quora’s algorithms address to me are either too silly, or too specialist and esoteric for my capabilities to respond.
In this case, after reading the other answers, I thought they had largely missed the mark, and perhaps the point. The authors may draw the same conclusion about my answer.
I found that my answer went in a subtly different direction to what I intended, but resulted in a mini-epiphany. There is a limit to what we can know because there will always be a limit to the mathematics we know, which thus far determines what we know of the cosmos.
My answer to What does logic reveal about reality?
Fundamentally, it reveals that there are limits to what we can know.
Epistemology is the ‘theory of knowledge’ (dictionary definition) - effectively, the study of what we can know. Whereas ontology is defined as ‘the nature of being’, which, in effect, is what we call reality.
Since the Enlightenment, it’s become increasingly apparent that it’s our knowledge of mathematics that determines the limits of what we can know, both at the cosmological and the infinitesimal scale. But mathematics itself has epistemological limits according to Godel’s Incompleteness Theorem.
In effect, Godel proved that, in any axiom based mathematical system, there will be mathematical truths that we can’t prove. In practice, this means that there will always be mathematical truths that lie beyond what we currently know. In this context, ‘what we currently know’ is transient. So even though we may, and will, know more in the future, it will never be complete.
The point is that we use logic to reveal these mathematical truths and so the corollary to Godel’s theorem is that there will always be a limit to what that logic can reveal, no matter how much it has revealed already. Basically, we extend our knowledge by extending our axiomatic system. To give an example: by employing the new axiom, √-1 = i, we uncovered a whole new realm of mathematics.
Some centuries later, we then used that particular mathematics (called complex algebra) to describe a newly discovered phenomena called quantum mechanics (QM). In fact, without that knowledge (revealed by pure logic) quantum mechanics would never have been developed into a consistent and highly successful theory. And arguably, QM is ‘the evanescent substrate on which we all exist’ [or reality] to quote Clifford A Pickover.
And this is the loop: QM is the substrate of the Universe, which created humans which discovered an abstract mathematics, which not only describes, but prescribes the rules for QM.
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