Mathematical Meditations

by Thomas Scarborough

I shall call this post an exploration—a survey. Is mathematics, as Galileo Galilei described it, ‘the language in which God has written the universe’? Are the numerical features of the world, in the words of the authors of the Collins Dictionary of Philosophy, Godfrey Vesey and Paul Foulkes, ‘free from the inaccuracies we meet in other fields'? Many would say yes—however, there are things which give us pause for thought.

• Sometimes reality may be too complex for our mathematics to apply. It is impossible to calculate in advance something as simple as the trail of a snail on a wall. Stephen Hawking noted, 'Even if we do achieve a complete unified theory, we shall not be able to make detailed predictions in any but the simplest situations.' If we do try to do so, therefore, we abuse mathematics—or perhaps we should say, in many contexts, mathematics fails. 

• Our measurement of the world may be inadequate to the task—in varying degrees. I take a ruler, and draw a line precisely 100 mm in length. But now I notice the grain of the paper, that my pencil mark is indistinct, and that the ruler's notches are crude. In many cases, mathematics is not the finest fit with the reality we deal with. In some cases, no fit at all. I measure the position of a particle, only to find that theoretical physicist Werner Heisenberg was right: I have lost its velocity. 

• The cosmologist Rodney Holder notes that, with regard to numbers—all numbers—'a finite number of decimal places constitutes an error'. Owners of early Sinclair calculators, such as myself, viewed the propagation of errors in these devices with astonishment. While calculators are now much refined, the problem is still there, and always will be. This error, writes Holder, 'propagates so rapidly that prediction is impossible'. 

• In 1931, the mathematician Kurt Gödel presented his incompleteness theorems. Numbers systems, he showed, have limits of provability. We cannot unite what is provable with what is true—given that what this really means is, in the words of Natalie Wolchover, ‘ill-understood’. A better known consequence of this is that no program can find all the viruses on one’s computer. Consider also that no system in itself can prove one’s own veracity. 

• Then, it is we ourselves who decide what makes up each unit of mathematics. A unit may be one atom, one litre of water, or one summer. But it is not that simple. Albert Einstein noted that a unit 'singles out a complex from nature'. Units may represent clouds with noses, ants which fall off a wall, names which start with a 'J', and so on. How suitable are our units, in each case, for manipulation with mathematics?

• Worst of all, there are always things which lie beyond our equations. Whenever we scope a system, in the words of philosophy professor Simon Blackburn, there is 'the selection of particular facts as the essential ones'. We must first define a system’s boundaries. We must choose what it will include and what not.  This is practically impossible, for the reason that, in the words of Thomas Berry, an Earth historian, 'nothing is completely itself without everything else'. 

• I shall add, myself, a 'post-Gödelian' theorem. Any and every mathematical equation assumes that it represents totality. In the simple equation x + y = z, there is nothing beyond z. As human beings, we can see that many things lie outside z, but if the equation could speak, it would know nothing of it. z revolts against the world, because the equation assumes a unitary result, which treats itself as the whole.
  

Certainly, we can calculate things with such stunning accuracy today that we can send a probe to land on a distant planet’s moon (Titan), to send back moving pictures. We have done even more wonderful things since, with ever increasing precision. Yet still the equations occupy their own totality. Everything else is banished. At what cost?

Picture Post #65 The Cell

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'Because things don’t appear to be the known thing; they aren’t what they seemed to be

neither will they become what they might appear to become.'

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Posted by Martin Cohen

‘Cellular landscape cross-section through a eukaryotic cell’
by Evan Ingersoll and Gael McGill. 

I was struck by the artificial, even ‘mathematical’ nature of this image, which is, on the contrary, a glimpse into something entirely natural and, if it is mathematical, it is a very strange kind of mathematics. It is in fact, a human cell at some fabulous magnification (maybe the colours have been added). It is, in other words, something both quite natural and yet completely unnatural – for human beings were never supposed to see such details. Or were we? There the philosophers might wrangle…

For what it's worth, the creators of the image used “X-ray, nuclear magnetic resonance, and cryo-electron microscopy datasets” for all of its “molecular actors”. And it is apparently less complex than a real cell. And one other detail is interesting about the image: it was inspired by the stunning art of David Goodsell, an Associate Professor in the Department of Integrative Structural and Computational Biology, where he says that he currently divides his time between research and science outreach… the outreach centred on the power of these other-worldly images.

Poetry: The Mathematical State of Love

 

Posted by Chengde Chen *

Some say love is mathematically positive
Like the state of ‘having’
Because only those who have can give
Man can love because he has feelings
God can love because He has power

Some say love is mathematically negative
Like the state of ‘owing’
The deeper one loves, the more one owes
Hence parents’ willing and uncomplaining
And lovers’ risking death for one another

In fact, the mathematical state of love is zero
When you are not giving, it doesn’t exist
When you are giving, it doesn’t decrease
Whether by multiplication or by division
It turns what is not into itself

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* Chengde Chen is the author of the philosophical poems collection: Five Themes of Today, Open Gate Press, London. chengde.chen@hotmail.com

Is Mathematics Invented or Discovered?

Posted by Keith Tidman

I’m a Platonist. Well, at least insofar as how mathematics is presumed ‘discovered’ and, in its being so, serves as the basis of reality. Mathematics, as the mother tongue of the sciences, is about how, on one important epistemological level, humankind seeks to understand the universe. To put this into context, the American physicist Eugene Wigner published a paper in 1960 whose title even referred to the ‘unreasonable effectiveness’ of mathematics, before trying to explain why it might be so. His English contemporary, Paul Dirac, dared to go a step farther, declaring, in a phrase with a theological and celestial ring, that ‘God used beautiful mathematics in creating the world’. All of which leads us to this consequential question: Is mathematics invented or discovered, and does mathematics underpin universal reality?

‘In every department of physical science, there is only so much science … as there is mathematics’ — Immanuel Kant

If mathematics is simply a tool of humanity that happens to align with and helps to describe the natural laws and organisation of the universe, then one might say that mathematics is invented. As such, math is an abstraction that reduces to mental constructs, expressed through globally agreed-upon symbols, serving (in the complex realm of human cognition and imagination) as an expression of our reasoning and logic, in order to better grasp the natural world. According to this ‘anti-realist’ school of thought, it is through our probing that we observe the universe and that we then build mathematical formulae in order to describe what we see. Isaac Newton, for example, developed calculus to explain such things as the acceleration of objects and planetary orbits. Mathematicians sometimes refine their formulae later, to increasingly conform to what scientists learn about the universe over time. Another way to put it is that anti-realist theory is saying that without humankind around, mathematics would not exist, either. Yet, the flaw in this scientific paradigm is that it leaves the foundation of reality unstated. It doesn’t meet Galileo’s incisive and ponderable observation that:

‘The book of nature is written in the language of mathematics.’

If, however, mathematics is regarded as the unshakably fundamental basis of the universe — whereby it acts, so to speak, as the native language of everything (embodying universal truths) — then humainity’s role becomes to discover the underlying numbers, equations, and axioms. According to this view, mathematics is intrinsic to nature and provides the building blocks — both proximate and ultimate — of the entire universe. An example consists of that part of the mathematics of Einstein’s theory of general relativity predicting the existence of ‘gravitational waves’, the presence of which would not be proven empirically until this century, through advanced technology and techniques. Per this kind of ‘Platonic’ school of thought, the numbers and relationships associated with mathematics would nonetheless still exist, describing phenomena and governing how they interrelate and in so doing bring a semblance of order to the universe — a universe that would exist even absent humankind. After all, this underlying mathematics existed before humans arrived upon the scene — merely awaiting our discovery — and this mathematics will persist long after us.

If this Platonic theory is the correct way to look at reality, as I believe it is, then it’s worth taking the issue to the next level, which is the unique role of mathematics in formulating truth and serving as the underlying reality of the universe — both quantitative and qualitative. As Aristotle summed it up,  the ‘principles of mathematics are the principles of all things’ — thus foreshadowing the possibility of what later became known in the mathematical and science world as a ‘theory of everything’, unifying all forces, including the still-defiant unification of quantum mechanics and relativity. 

As the Swedish-American cosmologist Max Tegmark provocatively put it, ‘There is only mathematics; that is all that exists’ — an unmistakably monist perspective. He colorfully goes on:

‘We all live in a gigantic mathematical object — one that’s more elaborate than a dodecahedron, and probably also more complex than objects with intimidating names such as Calabi-Yau manifolds, tensor bundles and Hilbert spaces, which appear in today’s most advanced physics theories. Everything in our world is purely mathematical— including you.’

The point is that mathematics doesn’t just provide ‘models’ of physical, qualitative, and relational reality; as Descartes suspected centuries ago, mathematics is reality.

Mathematics thus doesn’t care, if you will, what one might ‘believe’; it dispassionately performs its substratum role, regardless. The more we discover the universe’s mathematical basis, the more we build on an increasingly robust, accurate understanding of universal truths and, even, get ever nearer to an uncannily precise, clear window onto all reality — foundational to the universe. 

In this role, mathematics has enormous predictive capabilities that pave the way to its inexhaustibly revealing reality. An example is the mathematical hypothesis stating that a particular fundamental particle exists whose field is responsible for the existence of mass. The particle was theoretically predicted, in mathematical form, in the 1960s by British physicist Peter Higgs. Existence of the particle — named the Higgs boson — was confirmed by tests some fifty-plus years later. Likewise, Fermat’s famous last theorem, conjectured in 1637, was not proven mathematically until some 360 years later, in 1994 — yet the ‘truth value’ of the theorem nonetheless existed all along.

Underlying this discussion is the unsurprising observation by the early-20th-century philosopher Edmund Husserl, who noted, in understated fashion, that ‘Experience by itself is not science’ — while elsewhere referring to ‘the profusion of insights’ that could be obtained from mathematical research. That process is one of discovery. Discovery, that is, of things that are true, even if we had not hitherto known them to be so. The ‘profusion of insights’ obtained in that mathematical manner renders a method that is complete and consistent enough to direct us to a category of understanding whereby all reality is mathematical reality.

Is Mathematics Invented or Discovered?

Posted by Keith Tidman

I’m a Platonist. Well, at least insofar as how mathematics is presumed ‘discovered’ and, in its being so, serves as the basis of reality. Mathematics, as the mother tongue of the sciences, is about how, on one important epistemological level, humankind seeks to understand the universe. To put this into context, the American physicist Eugene Wigner published a paper in 1960 whose title even referred to the ‘unreasonable effectiveness’ of mathematics, before trying to explain why it might be so. His English contemporary, Paul Dirac, dared to go a step farther, declaring, in a phrase with a theological and celestial ring, that ‘God used beautiful mathematics in creating the world’. All of which leads us to this consequential question: Is mathematics invented or discovered, and does mathematics underpin universal reality?

‘In every department of physical science, there is only so much science … as there is mathematics’ — Immanuel Kant

If mathematics is simply a tool of humanity that happens to align with and helps to describe the natural laws and organisation of the universe, then one might say that mathematics is invented. As such, math is an abstraction that reduces to mental constructs, expressed through globally agreed-upon symbols. In this capacity, these constructs serve — in the complex realm of human cognition and imagination — as a convenient expression of our reasoning and logic, to better grasp the natural world. According to this ‘anti-realist’ school of thought, it is through our probing that we observe the universe and that we then build mathematical formulae in order to describe what we see. Isaac Newton, for example, developed calculus to explain such things as the acceleration of objects and planetary orbits. Mathematicians sometimes refine their formulae later, to increasingly conform to what scientists learn about the universe over time. Another way to put it is that anti-realist theory is saying that without humankind around, mathematics would not exist, either. Yet, the flaw in this paradigm is that it leaves the foundation of reality unstated. It doesn’t meet Galileo’s incisive and ponderable observation that:

‘The book of nature is written in the language of mathematics.’

If, however, mathematics is regarded as the unshakably fundamental basis of the universe — whereby it acts as the native language of everything (embodying universal truths) — then humanity’s role becomes to discover the underlying numbers, equations, and axioms. According to this view, mathematics is intrinsic to nature and provides the building blocks — both proximate and ultimate — of the entire universe. An example consists of that part of the mathematics of Einstein’s theory of general relativity predicting the existence of ‘gravitational waves’; the presence of these waves would not be proven empirically until this century, through advanced technology and techniques. Per this ‘Platonic’ school of thought, the numbers and relationships associated with mathematics would nonetheless still exist, describing phenomena and governing how they interrelate, bringing a semblance of order to the universe — a math-based universe that would exist even absent humankind. After all, this underlying mathematics existed before humans arrived upon the scene — awaiting our discovery — and this mathematics will persist long after us.

If this Platonic theory is the correct way to look at reality, as I believe it is, then it’s worth taking the issue to the next level: the unique role of mathematics in formulating truth and serving as the underlying reality of the universe — both quantitative and qualitative. As Aristotle summed it up, the ‘principles of mathematics are the principles of all things’. Aristotle’s broad stroke foreshadowed the possibility of what millennia later became known in the mathematical and science world as a ‘theory of everything’, unifying all forces, including the still-defiant unification of quantum mechanics and relativity. 

As the Swedish-American cosmologist Max Tegmark provocatively put it, ‘There is only mathematics; that is all that exists’ — an unmistakably monist perspective. He colorfully goes on:

‘We all live in a gigantic mathematical object — one that’s more elaborate than a dodecahedron, and probably also more complex than objects with intimidating names such as Calabi-Yau manifolds, tensor bundles and Hilbert spaces, which appear in today’s most advanced physics theories. Everything in our world is purely mathematical— including you.’

The point is that mathematics doesn’t just provide ‘models’ of physical, qualitative, and relational reality; as Descartes suspected centuries ago, mathematics is reality.

Mathematics thus doesn’t care, if you will, what one might ‘believe’; it dispassionately performs its substratum role, regardless. The more we discover the universe’s mathematical basis, the more we build on an increasingly robust, accurate understanding of universal truths, and get ever nearer to an uncannily precise, clear window onto all reality — foundational to the universe. 

In this role, mathematics has enormous predictive capabilities that pave the way to its inexhaustibly revealing reality. An example is the mathematical hypothesis stating that a particular fundamental particle exists whose field is responsible for the existence of mass. The particle was theoretically predicted, in mathematical form, in the 1960s by British physicist Peter Higgs. Existence of the particle — named the Higgs boson — was confirmed by tests some fifty-plus years later. Likewise, Fermat’s famous last theorem, conjectured in 1637, was not proven mathematically until some 360 years later, in 1994 — yet the ‘truth value’ of the theorem nonetheless existed all along.

Underlying this discussion is the unsurprising observation by the early-20th-century philosopher Edmund Husserl, who noted, in understated fashion, that ‘Experience by itself is not science’ — while elsewhere his referring to ‘the profusion of insights’ that could be obtained from mathematical research. That process is one of discovery. Discovery, that is, of things that are true, even if we had not hitherto known them to be so. The ‘profusion of insights’ obtained in that mathematical manner renders a method that is complete and consistent enough to direct us to a category of understanding whereby all reality is mathematical reality.

The Deceptions of Mathematics

by Thomas Scarborough

Galileo Galilei, a man of formidable scientific ability, once wrote that ‘the universe cannot be read until we have learned the (mathematical) language.’  Mathematics, he suggested, would reveal the secrets of the entire cosmos.  It is a common view – yet it is deceptive.  In fact, it may reveal little more than hubris.

On the surface of it, mathematics – even more than science – would seem to be thoroughly objective.  Here there are no failed experiments, no false interpretations, no paradigm shifts.  In mathematics – so it is frequently assumed – there is perfect certainty.

1 + 1 = 2
the logarithm of 1 = 0
the square root of 1 = 1
and so on.

Yet we overlook something, which would seem as simple as one-two-three.  We apply mathematics, by and large, to things in the real world (pure mathematics being the exception to the rule) – and in order so to apply it, we identify units of quantity.  This identification of units of quantity begins with ‘quantification’ – we map our human sense observations into units of quantity, or simply, quantity. 

On first impressions, it might seem to us then that units of quantity come ready made.  Apples come in ones, oranges come in ones – so do people, animals, days, nights, doors, windows, and a great deal more.  And if they do not come in ones, then we may make them into ones: one kilogram, one litre, one block, and so on.  On this basis, we quantify things and perform various mathematical operations on them. 

However, it is not this simple – and even a child might know it.  Our ‘ones’ may really be anything at all – say, clouds with noses (‘I saw three of them today’), ants which fall off the wall (dozens), or dogs which wag their tails, and so on to infinity.  In each case we are dealing with the mathematical unit ‘one’.

The theoretical physicist Albert Einstein would surely have agreed.  He considered that a unit ‘singles out a complex from nature’.  This surely seems a contradiction in terms.  A complex consists of many different and connected parts – parts (plural) which get ‘singled’, out.  That is, one takes a bundle of things or properties, and one defines them as one.  Therefore, various things and various properties may all at once hide inside one and the same single mathematical unit. 

Now this opens up an obvious question.  Who then is to say that our mathematical units – those complexes which we have ‘singled’, out – are precisely the complexes we need for the purpose of our calculations?  Supposing that we really ought to have added something to a complex which we call ‘one’ – or that we really should have taken something out – before we began to make use of it? 

Besides, does one really find such a thing as a complex which is self-contained and closed?  Is not every singled out complex-cum-unit criss-crossed by associations and influences without number?

When we think on it, this is true even of the simplest things in this world.  For instance, we might temporarily assume that the complex ‘hamster’ does not include food or water – it merely refers to a rodent, of which there are so and so many millions in the world.  Yet this complex breaks down at a certain point, as some children can tragically relate, who forgot the food or water.

Consider a thought experiment – as if it had never been conducted before.  Supposing it is true that our complexes might leave things out – or squeeze things in that really ought to be left out.  What then would the logical consequences be?  Of course, high on the list would be that our mathematics may not fit reality, because our mathematical units are ‘not quite right’.  Not only that, but we should easily find examples of this in the world. 

And so it is.  The mathematics of circular orbits and epicycles had to be replaced with the mathematics of elliptical orbits – the mathematics of scalars, then vectors, had to be replaced by the mathematics of tensors.  The mathematics of classical thermodynamics had to be replaced by the mathematics of generalised thermodynamics – and so on.  In fact our complexes may contain an entire world-view which needs to be overhauled – for example, Newtonian physics.  Yet even with the new, we would do well to remember that we have now carved up our world into four mathematical models.

The nineteenth century American philosopher Charles Sanders Peirce saw that ‘every new concept first comes to the mind in a judgment.’  He was saying, apparently, that our ‘ones’ are simply creations of the mind.

On this basis, we may assume that even the simplest of mathematics is not as straightforward as it seems.  In fact mathematics, writes the pioneering statistician William Briggs, requires ‘slow, maturing thought’.  It is not just about numbers, but about wisdom and expansive thinking.

The deceptions are, therefore, that mathematics is objective – and that being objective, it makes an excellent fit with our world – perhaps a perfect fit with the cosmos, as Galileo suggested.   No.  On the contrary, we should see mathematics as a very flawed and very subjective tool – always too simplistic, always in some way violating the totality of the reality in which we live. Mathematics, at the least, should be handled with great humility.

What Are ‘Facts’?

On the trail of the Higgs Boson

Posted by Keith Tidman

What are 'facts'? The ages-long history of deception and sleights of hand and mind — including propaganda and political and psychological legerdemain — demonstrates just one of the many applications of false facts. But similar presentations of falsities meant to deceive, sow discord, or distract have been even more rife today, via the handiness and global ubiquity of the Internet. An enabler is the too-frequent lack of judicious curation and vetting of facts. And, in the process of democratizing access to facts, self-serving individuals may take advantage of those consumers of information who are ill-equipped or disinclined (unmotivated) to discern whether or not content is true. Spurious facts dot the Internet landscape, steering beliefs, driving confirmation bias, and conjuring tangible outcomes such as voting decisions. Interpretations of facts become all the more confounding in political arenas, where interpretations (the understanding) of facts among differently minded politicians becomes muddled, and ‘what’s actually the case’ remains opaque.

And yet surely it is the total anthology of facts — meaning things (their properties), concepts, and their interrelationships — that composes reality. Facts have multiple dimensions, including what one knows (epistemological aspects), how one semantically describes what’s known (linguistic aspects), and what meaning and purpose one attributes to what’s known (metaphysical aspects).

Facts are known on a sliding scale of certainty. An example that seems compelling to me comes from just a few years ago, when scientists announced that they had confirmed the existence of the Higgs boson, whose field generates mass through its interaction with other particles. The Higgs’s existence had been postulated earlier in mathematical terms, but empirical evidence was tantalizingly sought over a few decades. The ultimate confirmation was given a certainty of ‘five sigma’: that there was less than 1 chance in 3.5 million that what was detected was instead a random fluctuation. Impressive enough from an empirical standpoint to conclude discovery (a fact), yet still short of absolute certainty. With resort to empiricism, there is no case where some measure of doubt (of a counterfactual), no matter how infinitesimally small, is excluded.

Mathematics, meantime, provides an even higher level of certainty (rigor of method and of results) in applying facts to describe reality: Newtonian, Einsteinian, quantum theoretical, and other models of scientific realism. Indeed, mathematics, in its precise syntax, universal vocabulary, and singular purpose, is sometimes referred to as the language of reality. Indeed, as opposed to the world’s many natural languages (whose known shortcomings limit understanding), mathematics is the best, and sometimes the only, language for describing select facts of science (mathematical Platonism) — whereby mathematics is less invented than it is discovered as a special case of realism.

Facts are also contingent. Consider another example from science: Immediately following the singularity of the Big Bang, an inflationary period occurred (lasting a tiny fraction of a second). During that inflationary period, the universe — that is, the edges of space-time (not the things within space-time) — expanded faster than the speed of light, resulting in the first step toward the cosmos’s eventual lumpiness, in the form of galaxies, stars, planets. The laws — that is, the facts — of physics were different during the inflation than what scientists are familiar with today — today’s laws of physics breaking down as one looks back closer and closer to the singularity. In this cosmological paradigm, facts are contingent on the peculiar circumstances of the inflationary epoch. This realization points broadly to something capable of being a fact even if we don’t fully understand it.

The sliding scale of certainty and facts’ contingency apply all the more acutely when venturing into other fields. Specifically, the recording of historical events, personages, and ideas, no matter the scholarly intent, often contain biases — judgments, symbols, interpretations — brought to the page by those historians whose contemporaneous accounts may be tailored to self-serving purposes, tilting facts and analyses. In natural course, follow-on historians inadvertently adopt those original biases while not uncommonly folding in their own. Add to this mix the dynamic, complex, and unpredictable (chaotic) nature of human affairs, and the result is all the more shambolic. The accretion of biases over the decades, centuries, and millennia doesn’t of course change reality as such— what happened historically has an underlying matter-of-factness, even if it lingers between hard and impossible to tease out. But the accretion does distort (and on occasion even falsify) what’s understood.

This latter point suggests that what’s a fact and what’s true might either intersect or diverge; nothing excludes either possibility. That is, facts may be true (describe reality) or false (don’t describe reality), depending on their content. (Fairies don’t exist in physical form — in that sense, are false — but do exist nonetheless, legendarily woven into elaborate cultural lore — and in that sense, are true.) What’s true or false will always necessitate the presence of facts, to aid determinations about truth-values. Whereas facts simply stand out there: entirely indifferent to what’s true or false, or what’s believed or known, or what’s formally proven, or what’s wanted and sought after, or what’s observable. That is, absent litmus tests of verifiability. In this sense, given that facts don’t necessarily have to be about something that exists, ‘facts’ and ‘statements’ serve interchangeably.

Facts’ contingency also hinges in some measured, relativistic way on culture. Not as a universally  normative standard for all facts or for all that’s true, of course, but in ways that matter and give shared purpose to citizens of a particular society. Acknowledged facts as to core values — good versus evil, spirituality, integrity, humanitarianism, honesty, trustworthiness, love, environmental stewardship, fairness, justice, and so forth — often become rooted in society. Accordingly, not everyone’s facts are everyone else’s: facts are shaped and shaded both by society and by the individual. The result is the culture-specific normalising of values — what one ‘ought’ to do, ideally. As such, there is no fact-value dilemma. In this vein, values don’t have to be objective to be factual — foundational beliefs, for example, suffice. Facts related to moral realism, unlike scientific and mathematical realism, have to be invented; they’re not discoverable as already-existing phenomena.

Facts are indispensable to describing reality, in both its idealistic (abstract) and realistic (physical) forms. There is no single, exclusive way to define facts; rationalism, empiricism, and idealism all pertain. Yet subsets of facts, and their multifaceted relationships that intricately bear on each other’s truth or falsity, enable knowledge and meaning (purpose) to emerge — an understanding, however imperfect, of slices of abstract and physical reality that our minds piece together as a mosaic. 

In short, the complete anthology of facts relates to all possible forms of reality, ranging the breadth of possibilities, from figments to suppositions to the verifiable phenomenal world.