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Where numbers come from

June 6, 2022

Of course it all depends on what numbers are taken to be. Numbers are not real. You cannot see or touch or smell them. They are labels (words) with associated symbols (numerals). They are neither nouns nor adjectives though, in some contexts, they can be used as nouns. Philosophy calls them abstract objects. They are perceived as abstractions in the real world of existence. But they are abstractions which display relationships and patterns among themselves quite independent of the applications from which they are discerned. What lies at the source of numbers? How did they come to be? I see numbers as a means, a language, for describing the underlying patterns connecting countable things in the existing universe.

It starts with one. 

There are four abstract concepts, I suggest, which lie at the source not only of all numbers and numbering systems but of mathematics in general (beginning with arithmetic and geometry). It seems to me that these four concepts are necessary and sufficient.

    1. Oneness (1)
    2. Set
    3. Sum
    4. Arithmetical addition (+)

If there were nothing to count, we would not have numbers.

Of things that exist, even abstract things, human cognition distinguishes between countable things and uncountable things. The necessary and required conditions for things to be considered countable are, first that they exist, and second that each has a unique, discrete identity. Things that are uncountable are those perceived as being part of a continuum and therefore not discrete or unique. (The word uncountable is also used for countable things which are too numerous for the human mind to contemplate but it is not that meaning that I use here). Thus apples, oranges, grains of sand, people and atoms are all countable. Things and concepts that cannot be divided into discrete elements are uncountable. Space, sky, water, air, fire, and earth are all perceptions of continua which are uncountable. Nouns for generalisations are uncountable (furniture, music, art, ….). The distinction applies to abstract things as well. Discrete thoughts or specific ideas can be counted. But abstractions (shapes, information, news, advice, ….) which lack a discrete, unique identity are within the ranks of the uncountable. Types of emotions are countable, but emotion is not.

To be discrete and unique give substance to identity. Existence (a Great Mystery) comes first, of course. To have identity is to have some distinguishing characteristic which enables the quality of “oneness”. Note that the quality of being identical (similar) does not disturb identity. Two, or many, things may be identical, but the identity of each remains inviolate. An atom of hydrogen here may be identical to an atom of hydrogen elsewhere, but the identity of each remains undisturbed. It is estimated that there are between 1078 to 1082 atoms existing in the observable universe. Each one distinct from all the others. Each one having identity.

We use the word identity in many contexts. In the philosophical sense, which includes the context of counting, my definition of identity is then:

identity – oneness; the distinguishing character of a thing that constitutes the objective reality of that thing

It is the discreteness and uniqueness contained in identity which gives rise to the concept of oneness as a quality of a thing which makes that thing countable. It is having the concept of oneness which allows us to define a concept of number, label it as “one” and give it a symbol (1). How the concept of identity (oneness) emerged in the species is another one of the Great Mysteries of life and consciousness.

But the concept of identity alone is not sufficient to generate the need to count and the invention of numbers. Having defined one (1), something else is still needed to generate a number system. A social, behavioural component is also required; It is cooperation and interaction with others which leads to the need to count. It probably emerged when humans created social groupings and things were accumulated for rainy days. The notion of addition as the accumulation of things belonging to a set is also needed. An ancient human may have gathered an apple an orange and a goat and accumulated many things but would probably not have thought of those things as belonging to the set of things. If he had gathered only apples and oranges, he may well have recognised that he had accumulated a set of things identified as fruit. And someone at sometime in our prehistory did note that his accumulation of individual goats all belonged to the set of things identified as goats. We cannot now know how our ancestors first came to a numbering system and the concept of addition with numbers, but it must certainly have been at around the same time that the need for counting emerged.

To get from just observing the accumulation of things in the real world to the concept of arithmetical addition was a major intellectual leap.  That journey also needed that the concepts of a set and of a sum were in place. We can only speculate on how that emergence and conjunction of the necessary concepts took place. It would surely have been noticed that there was a common, underlying pattern (rule) which applied with the accumulation – separately – of, say, apples and / or goats. But it would also have been noticed that the pattern did not apply when dealing with mixtures of apples and goats together. Accumulating an apple and an apple exhibits the same underlying pattern as accumulating a goat and a goat. But a goat and an apple followed the same rule only when they were considered, not as goats or apples, but as things belonging to a greater class (set) of things.

1 apple + 1 apple follows the same abstract, underlying pattern as 1 goat + 1 goat or 1 thing + 1 thing, but the rule breaks down at 1 apple + 1 goat.

A set of things – is a multiplicity of similar countable things which together can assume a separate identity (unique and discrete)

It is likely that it was then that they realised that the accumulation of things could be represented by abstract rules (patterns) which were independent of the set of things being accumulated. The rule of arithmetical addition (+), they would have found, applied in a like manner to accumulations of members of any set, and that a common name could be given to the result (sum) of the accumulation.

Sum – the result of an accumulation

But they would also have found that the rule (pattern) of accumulation and counting broke down when dealing with mixed sets of things. Whereas one apple and one apple gave the same sum as one goat and one goat, that sum had no meaning if one apple was accumulated with one goat. However, the summation rule reasserts itself when considering the sum of things with the accumulation of one thing (apple) and one thing (goat). This general, but abstract, rule of the summation operation was arithmetical addition.

Arithmetical addition (+) – the accumulation of one number to another number giving a sum as the result

Maintaining identity remained crucial. They would have noted that the abstract rule did not apply if the things being accumulated lost identity (their oneness) during the operation. One goat and one lion gave one lion. One bubble and one bubble could merge to give one bubble. But they would also have noted that uncountable things were not capable of being accumulated.

Given the abstract concepts of identity (oneness, 1) and arithmetical addition (+), all natural numbers inevitably follow. With a 1 and with a +, and the concept of a set and a sum, all the natural numbers can be generated.

1 + 1 + 1 + 1 ……

Having invented a label and a symbol for oneness (one, 1), new labels and symbols were then invented for the abstract sums. The chosen base (binary, decimal, hexagesimal, vigesimal, ….) determines the number of labels and symbols to be invented.

1 + 1 gave 2, 1+ 2 gave 3, ….. and so on

And all the natural numbers were born.

The reverse of accumulation, the giving away or lessening of things, led to the abstraction of arithmetical subtraction (-) and that gave us zero and all the negative integers. Note that oneness and one (1) must come first before the concept of zero (0) makes any sense. Zero is a very specific quality of nothingness, but zero is not nothingness. In the observed world an absence of apples (0 apples) is not quite the same thing as an absence of goats (0 goats), but the number abstraction (0) from both is the same. As a number, zero is special and does not follow all the rules discovered connecting, and applying to, the other numbers. (Zero also functions as a placeholder in our notations but that is a different matter to its properties as a number). Zero added to another number does not create a new number as every other number does. Division is allowed by any number but not by zero. Division by zero is undefined. One (1), not zero (0), is where numbers start from. Zero is merely a consequence of removing (subtracting) one (1) from one (1).

Multiplication is just recursive addition. Recursive subtraction leads to division and that generates irrational numbers. Applying numbers to shapes (geometry) led to the discovery of transcendental numbers. Number theory is about studying and discovering relationships between numbers. But all these discovered relationships exist only because numbers exist as they do. All the relationships exist as soon as the concepts of oneness (1) and addition (+) are fixed together with the concepts of a set and a sum. Discoveries of the relationships can come much later. Numbers depend on counting and number theory depends upon the numbers.

Numbers start with one (1), and without a one (1) there can be no numbers.

Numbers, ultimately, rest on the concept of identity (oneness).

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When the tree falls in the forest, the sound is only due to language

May 1, 2022

The classic, cliched question goes:

“If a tree falls in a forest and no one is around to hear it, does it make a sound?”

The non-philosophical part of question is easily answered.

  • If a tree falls within an extant medium, and
  • there is consequent vibration within that medium, and
  • there is an organ which can detect such vibrations, and
  • the organ generates impulses, and
  • it sends these impulses to a brain, and
  • that brain interprets the impulses as something the brain itself labels as sound, then
  • there will be sound.

If the tree fell in a vacuum there would be no vibration of anything. No medium, no sound. No ear, no sound. No brain. no sound. In fact, if we did not have ears connected to our brains our language would be unable to come up with words for ears, hearing or sound. If we had no word for sound then there might well be vibrations when the tree fell, but there would be no sound. The non-philosophical answer then becomes that if we had no word in language for sound then there would be no sound. When a dog or a bat detects vibrations at frequencies that our ears cannot detect then such signals never reach our brains to ever be classified in our language as sound. What an animal might interpret in its brain when its ears detect signals is whatever that animal interprets it or labels it to be.  Only if we define the word sound to loosely mean what any brain may interpret on receiving signals from any ear-like organ, could we say that the animal discerns sound.

The philosophical part of the question, however, which considers perception, observation and existence is much more interesting. There are many things we cannot directly experience with our limited senses. But we can infer and/or deduce that they exist by their interactions with other things giving changes which we can observe directly. We extend our senses by creating wonderful instruments which then produce changes observable directly by our traditional senses. We “see” in the ultraviolet or the infra-red only because our cameras convert these UV or IR signals into images that do fall within our visible range.

But what of all that we cannot observe, directly or indirectly, by our limited senses and our finite brains? Is it so that if something cannot be observed, cannot be perceived, cannot be inferred to exist by any interaction it has with anything else in this universe, then it does not exist? Or is it merely that we are ignorant of its existence? Philosophy is, of course, about asking unanswerable questions. Once a question can be answered it leaves the field of philosophy.

Take bongism for example. We cannot observe it, perceive it, infer it or deduce it. It has no known interactions with anything else in this universe. But it is the imbalance in bongism which caused all existence in the first place. It is the answer to the question “Why do things exist at all?”.

Does bongism exist?

It must do, since I have a word for it.

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The subjectivity of objective

April 18, 2022

Absolute objectivity is a mirage. Objectivity, in itself, is always a value judgement and always subjective. There is no observation, no experience, no proposition, no fact, no truth, no logic which has not been filtered through human cognition and all its shortcomings. Nothing is completely objective. Nothing I write can be objective. That is a truth which comes before the beginning.

Let us start there.

Objective is not a useful word in framing an insult. “You objective scoundrel” somehow elevates a “scoundrel” and detracts from the insult. I cannot think of an example where being objective is considered bad. An objective evil or an objective crime are word combinations without meaning. At worst, objective is perceived as neutral. In regard to human thoughts and actions we assume that they are either based on logic and reason or on feelings and emotion. They are not necessarily opposed but it is implicit in our language that they are different. We perceive reasoning to be more objective than emotional reactions.

We allow the ability to distinguish objective from subjective to reside only in animate things having brains. We do not even allow artificial brains that ability. We know that brains are where both logic and reason on the one hand, and feelings and emotions on the other, reside. But we connect being objective with a brain’s exercise of logic and reason and untainted by emotions. Language does not permit an emotional rationality. Being subjective is also of a brain but is a characteristic of the individuality of that brain and its attendant emotions. Subjectivity is undefined without a brain which generates both reason and emotion. The practice of science and the law thus set a high value on the thing we call objectivity, whereas we appreciate, and expect, an individualistic subjectivity from an author or a musician or a painter or a teacher or a tennis champion.

But in seeking objectivity we are chasing a mirage.

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The innate goodness of liberals justifies their being authoritarian and even fascist

February 16, 2022

The fundamental truth is that a “liberal” is a member of a higher species, homo sapiens sapiens superieur.

To be a liberal is to be an enlightened, right-thinking, superior type of human being. The superiority is a moral and intellectual superiority which derives from the certainty of thinking correctly. Thus the liberal knows, without any shadow of doubt, that any opposing views, whether expressed openly or only hiding in the sewers of someone else’s mind, are wrong. The rightness of the liberal view is absolute and must be accepted as a brute fact of our universe. To challenge this is not just immoral, but evil. The innate goodness in a liberal is such that all thoughts and actions of a liberal are, per force, “good”. It follows that liberals must rule over non-liberals for the good of the non-liberals. Non-liberals must be forbidden from thinking or doing anything that may harm their own humanity. The use of coercive force to ensure that non-liberals follow the liberal rules – for their own good – is justified and inherently “good”. If necessary, any rotten non-liberals may be cancelled by liberals. The cancellation, in the extreme case, may consist of physical extermination. But it is a “good” extermination since exterminating a non-liberal is always a “good thing”.

For a member of a superior species, as any liberal is, members of lower species are to be taken care of for their own good, just as other pets are taken care of. A liberal does not indulge in unnecessary cruelty and only uses necessary force. Non-liberals are to be well treated and well trained and only culled as a last resort, and even then with great humanity. However the liberal – by definition – always acts responsibly. In exposed spaces, non-liberals are not to be allowed to run freely or destructively. They are to be kept on a leash whenever others are around. If there is a risk that they will say wrong things, they are to be muzzled. Non-liberals may be allowed to have a say in their own future but the liberal will exercise his veto if non-liberals are not thinking correctly.

A liberal enjoys an exalted state and has a place reserved in heaven. Nothing he can do can take that away from him. Normally, a reservation in heaven would be jeopardised by cruel and coercive behaviour to others. However, authoritarianism and fascism are perfectly allowable if one is first a liberal. Coercion with necessary force is permitted when applied to a lower species, such as non-liberals or pet dogs.

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Physics theories are remarkably similar to God theories

February 6, 2022

I was listening to lectures by Carlo Rovelli on Loop Quantum Gravity and Sean Carroll on Quantum Wave Theory. While the maths used in modern physics is beyond my capabilities, it is very evident that the leading physicists of today when propounding their theories do not sound so very different to priests talking about their gods.

WSU Master Class: Loop Quantum Gravity with Carlo Rovelli

Mysteries of Modern Physics by Sean Carroll

I take physics to be the all-encompassing field which includes, among other scientific disciplines, astrophysics, astronomy, cosmology, particle physics and quantum mechanics. In one critical sense physics lies at the base of all the physical sciences and thus chemistry must be a sub-set of physics (though no self-respecting chemist would ever openly admit that). Mathematics, of course, is a language (actually many languages) invented to describe the world around us. The more precise and specific a branch of mathematics the more esoteric its application. It appears, at first glance, that physics gives chemistry which gives rise to organic chemistry which, in turn, leads to biochemistry. However, there is a crucial element missing when considering biochemistry as merely a sub-set of chemistry and then of physics. Neither physics nor chemistry can explain how the spark of life which animates biochemistry and biology came to be or why it should be at all. Some other unknown thing, in addition to physics, is needed to convert chemistry into biochemistry and for living things to emerge. The brute empirical fact of the existence of life and living things becomes both a fundamental assumption and a boundary condition for biology.

There are no physics theories which do not start with some fundamental assumptions which generally make up the initial and boundary conditions for the field of study. The field of study does not, cannot, thereafter, penetrate why those assumptions must be. Physics assumes causality and therefore cannot conceive of any non-causal events. (A contradiction arises whenever physics relies upon or invokes a truly random event since such an event must be – by definition – without cause). Biological and medical sciences start with the assumption of the existence of living things and do not, thereafter, concern themselves with the “trivial” question of why life came to be. The scientific process, in every branch of science, assumes that all observations are explainable, that causality prevails, and that the flow of time is regular, one-directional and irreversible. Philosophy and metaphysics sometimes address existence and causality and the nature of time but no science and no logic can address the fundamental assumptions it is itself built upon. Assumptions enable the many fields of study but they also constrain the field of study.

At the level closest to metaphysics lies the Standard Model of Cosmology which, in turn, is built upon the Standard Model of Particle Physics and the General Theory of Relativity. They all need fundamental assumptions which the models themselves cannot address. It is when justifying or explaining these basic assumptions (beliefs) that physicists and cosmologists become indistinguishable from theologists justifying the existence of the Divine.

The current Standard Model of Cosmology (SMC), also called the “Concordance Cosmological Model” or the “ΛCDM Model,” assumes that the universe was created in the “Big Bang” from pure energy, and is now composed of about 5% ordinary matter, 27% dark matter, and 68% dark energy. While the SMC is based primarily upon two theoretical models:

  1. the Standard Model of Particle Physics (SMPP), which describes the physics of the very small in terms of quantum mechanics and
  2. the General Theory of Relativity (GTR), which describes the physics of the very large in terms of classical mechanics;

it also depends upon several additional assumptions. The main additional assumptions of the SMC are:

  1. the universe was created in the Big Bang from pure energy;
  2. the mass energy content of the universe is given by 5% ordinary matter, 27% dark matter, and 68% dark energy;
  3. the gravitational interactions between the above three components of the mass energy content of the universe are described by the GTR; and
  4. the universe is homogeneous and isotropic on sufficiently large (cosmic) scales.

I note that the certainty of our science is based on observation of just part of the 5% of the universe which is observable. The other 95% (presumed to be and labeled dark energy and dark matter) is not observable but is imbued with just those properties needed to make our observations of part of the 5% fit into the Standard Model of the whole. They are, in fact, fudge factors to make observations fit a model. The God of Fudge Factors is brought into play but God forbid that it be considered a God. Dark energy and dark matter are just labels for unknown, magical sources of undetectable, unobservable matter and energy inferred to exist. Dark energy and dark matter are as true, and as slippery as heaven and hell are in theology.

The universe of this Standard Model starts without space, without time, and without any laws governing what causality should be. Physics and cosmology cannot address the question of existence (an assumed initial condition) and therefore resorts to trickery to create something local from a global nothing.

(Net zero)global = (+ something)local + (-something)local-elsewhere

This trick allows matter and energy (locally) to be “created” from a global nothing. We cannot detect enough anti-matter to balance all the matter we observe in our local universe, but don’t worry, it must be out there somewhere else. But that is not all. Just as our ancient ancestors invoked gods when there was no explanation, modern physicists invoke random events happening entirely by chance – as just one probability of many, that just happened to realised. Truly random must be without cause. Anything without cause is remarkably magical. To assign divinity to the magical is just a small step. Whenever it is claimed that it was pure, probabilistic chance that led to our particular universe or parts of it coming into existence, it is no more than an invocation of the Supreme God of Random Events.

Where there is no energy, pure chance allowed the use of this trick such that a

net zero = +(any amount of energy) – (the same amount of energy),

This gives some positive energy (from nothing) while at the same time (what time?) creating an equal amount of negative energy somewhere else (what somewhere else without any space?). Just preceding the Big Bang, Cosmic Inflation started (why?) and created space which allowed the laws of physics to emerge. Time emerged (why?), all entangled with the space, and this all somehow led to the Big Bang. They are all just Creation stories. Listen to cosmologists talking about Cosmic Inflation, or about the appearance of a local positive energy when the net global energy is zero and it is like listening to theology.

Whenever a physicist today claims that something is emergent, it is because that something defies explanation. In fact all the various speculative theories trying to bring quantum theory and gravitation together (string theory, loop quantum gravity, asymptotically safe gravity, causal dynamical triangulation, and emergent gravity) are all theories ultimately about the existence of our universe. When quantum mechanics brings in Everett’s universal wavefunction which collapses to give everything that existed, exists or will exist, we have just reached a God of Wavefunctions which rules them all.  Listening to the avid proponent of any particular theory is not so different to listening to an incomprehensible Sufi mystic. Hearing a string theorist arguing against a loop quantum gravity adherent is very like listening to the noises made by a Sunni arguing against a Shia. 

We need to remember that all God and Physics theories ultimately originate from inexplicabilities. Every mystery allows room for an explanatory theory which can be labeled a god. The Great Mysteries, which in past times would have been couched in terms of the Divine and called theology, are today couched in the language of mathematics and called Physics.

I seem to go around in circles but I can reach no other conclusion than that Gods and Physics theories are both just attempts to explain the inexplicable.

Science needs its Gods and religion is just politics

This essay has grown from the notes of an after-dinner talk I gave last year. As I recall it was just a 20 minute talk but making sense of my old notes led to this somewhat expanded essay. The theme, however, is true to the talk. The surrounding world is one of magic and mystery. And no amount of Science can deny the magic.

Anybody’s true belief or non-belief is a personal peculiarity, an exercise of mind and unobjectionable. I do not believe that true beliefs can be imposed from without. Imposition requires some level of coercion and what is produced can never be true belief. My disbelief can never disprove somebody else’s belief.

Disbelieving a belief brings us to zero – a null state. Disbelieving a belief (which by definition is the acceptance of a proposition which cannot be proved or disproved) brings us back to the null state of having no belief. It does not prove the negation of a belief.

[ (+G) – (+G) = 0, not (~G) ]

Of course Pooh puts it much better.

Science needs its Gods and religion is just politics

science-needs-its-godsDownload
https://ktwop.files.wordpress.com/2021/04/science-needs-its-gods.pdf

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Discovery versus invention and why mathematics is just another language

October 31, 2021

Languages are invented, and words are invented. We can assign whatever meaning we can commonly agree on, to any word. Discovery and invention are such words and are labels, placeholders, for some intended meaning. Confusion with words arise because the nuances of meaning associated with a word may not be as common as we think. This post is about mathematics being a language, but I need to start by what I understand to be discovery and how it differs from invention.

Discoveries and inventions

I do not quite understand the uncertainty sometimes expressed about whether things are inventions or discoveries. I find no ambiguity between “discovery” and “invention”. It is not as if everything in the world must be either discovered or invented. There is no need for any epistemological issues here. Discovery consists of the first finding of that which exists but is unknown. It includes what may be newly inferred either by deduction or by induction. What has been discovered by others can still be discovered by someone else as new knowledge. The discoverer, however, does not impact what is discovered. Discovery is always about finding. Invention, on the other hand, is always about creation. It creates something (material or immaterial) that did not exist before. The act of invention requires purpose and always results in a construct. The inventor defines and creates the invention. Once invented by someone, something may be discovered by others. An invention may be so complex that even its inventor has to discover some of its detailed characteristics or workings. It may be rediscovered if it has been forgotten (if immaterial) or has fallen into disuse (if material). Creation is not necessarily or always invention. A copy or even an improvement of an invention by someone else may involve creation but is neither a discovery nor an invention. Both finding and creation imply a doer, though it is not necessary for the doer to be human.

Of course everything discovered to exist must – so our logic tells us – have had a beginning. Whether such a beginning was a creation event (an invention), or random happenchance, or something else, lies at the heart of the most intractable philosophical question of all – the mystery of existence. I take the view that existence encompasses both material and immaterial things, and that the immaterial does not necessarily require a mind within which it is perceived to exist. The flow of time, causality and the laws of nature are, to my mind, examples of such immaterial existence. Thoughts, however, are also immaterial and exist but they need a mind within which to reside.

  • we discover our existence, we invent our names
  • emotions are discovered, their descriptive names are invented
  • hate is discovered, war is invented
  • a thought is discovered, a story is invented
  • the laws of nature are discovered, the laws of man are invented
  • knowledge is discovered, the Koran and the Bible are invented
  • human capabilities are discovered, actual behaviour can often be invented
  • the cognitive ability to relate and recognise patterns in sound is discovered, a piece of music is invented
  • language ability is discovered, languages are invented
  • new individual experiences (emotions, colours, feelings) are discovered, naming them may be inventions
  • individual learning of what is knowable but not known is discovery, creating a new word is invention
  • (Pooh learning about elephants was an invented discovery, naming them heffalumps was invention by Milne)
  • the cognitive ability to tell lies is discovered, fiction is invented
  • the human need to seek explanations is discovered, gods are invented
  • the emotional, human need for spiritualism is discovered, religions are invented
  • logic is discovered, argument is invented
  • relationships and patterns in the universe are discovered, languages to describe these are invented
  • where no man has gone before leads to discovery, naming or mapping the where is invention
  • my need for coffee is a discovery, my mug of coffee is a creation but is no invention
  • human behaviour may be discovered, “social science” is always invention
  • new geography is usually discovery and no geography can ever be invented
  • the past is immutable, history is a patchwork of discovered islands in a sea of invented narrative
  • news is discovered, fake news is invented

As a generalisation there is scientific discovery and there is artistic invention. The process of science is one of discovery and there is little ambiguity about that. The search for knowledge is about discovery and never about the invention of knowledge. That tools and instruments used for this process of discovery are often inventions is also apparent. Some ambiguity can be introduced – though I think unnecessarily – by treating postulates and hypotheses and theories as inventions in their own right. Thoughts are discovered, never invented. Any hypothesis or theory – before being put to the test – is a discovered thought, whether based on observations or not. Theories are discovered thoughts, but a conspiracy is always invented. Of course, observation (empiricism) may lead to conjecture, which may then take the form of a postulate or a theory. But there is no need for such conjectural thought to be equated with invention (though it often is).  Science discovers, engineering often invents. A good scientist discovers and uses inventions to further his discoveries. A good scientist may also be, and often is, an inventor. (Of course a great many calling themselves scientists are just bean counters and neither discover nor invent). Composers and authors and lawyers often invent. Doctors discover a patient’s ailments, a quack invents them. Most pharmaceutical companies invent drugs to suit discovered illnesses, some less ethical ones invent illnesses to suit their discovered compounds. Talent is discovered, a celebrity is invented.

Then there is the question of whether mathematics is discovered or invented. But before that we must define what it is that we are considering.

What mathematics is

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My morning coffee, free will and the mind-body problem

October 10, 2021

All the unknown laws of nature

It is early in the morning, and I am having my first mug of coffee today. I wonder how it arrived on the desk in front of me. Perhaps this mug of coffee today resulted unavoidably, inevitably, from the causal chain of events following a collision between two atoms 13.7 billion years ago. In a hard, deterministic world, it could not have arrived consequent to any purpose, for in such a world there can be no purpose. So, it could not possibly be that it was a conscious, abstract thought in my mind which initiated a variety of otherwise unrelated actions which led to this mug of coffee before me now. In such a world, it could not be that it was my abstract purpose alone that initiated the otherwise unrelated actions necessary to that purpose. My thoughts and perceived purpose could, in a material world, only be illusions. Perhaps if Genghis Khan had been born a day earlier, I would be having a masala chai with yak’s milk instead.

Or perhaps, just perhaps, it was in fact my free will, expressed as an abstract purpose, which, somehow, initiated and directed diverse interactions in the material world which resulted in my mug of coffee.

The “laws of nature” we acknowledge are restricted to the material world. They govern the behaviour of everything material (matter, energy, fields, waves, “dark” things, “strings” and even space). Interactions between the immaterial and the material are denied and therefore not subject to the laws of nature. This is a trifle self-contradictory since the laws themselves are abstract and immaterial. To claim that the laws emerge from the material world itself is somewhat specious since it requires the material world to be causeless and omniscient. It invokes simultaneously the God of Existence, his father the God of Random Events, and his uncle the God of Causality. Moreover it does not make the laws any less abstract. What then could be the unknown, causal, laws – also presumably laws of nature – which enable abstractions to cause actions in the material world and deliver my morning mug of coffee?

Wider than the mind-body problem

The mind-body problem has been a matter for philosophers for at least 2,500 years from pre-Aristotelian times.  It’s modern formulation can be traced to the writings of René Descartes, and especially the letters written to him by Princess Elizabeth of Bohemia between 1643 and 1650. She was a remarkable woman from a remarkable family, and it is her formulation which so clearly describes the issue recognised as the mind-body problem today.

She wrote, 370 years ago:

….. how the human soul can determine the movement of the animal spirits in the body so as to perform voluntary acts—being as it is merely a conscious substance. For the determination of movement seems always to come about from the moving body’s being propelled—to depend on the kind of impulse it gets from what sets it in motion, or again, on the nature and shape of this latter thing’s surface. Now the first two conditions involve contact, and the third involves that the impelling thing has extension; but you utterly exclude extension from your notion of soul, and contact seems to me incompatible with a thing’s being immaterial…

Nowadays the mind-body problem is seen as the unknown relationship between physical properties and mental properties. It is often broken down further into component parts: (more…)

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Where and when we are – a dimensional conundrum

September 7, 2021

“In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it”. – Wikipedia

In the concept of spacetime one might think that (x,y,z,t) are the four dimensional coordinates which are necessary and sufficient to specify the location of any object at any time within our universe. But that would be an oversimplification. It is true only for a relative location and not for any absolute location.

In reality we have no idea – in absolute terms – of where we are or when we are.

The place where I was born on the surface of the Earth has – during my lifetime – drifted along with the continents some 2.3 m North-East across the earth’s surface. The Sun (along with the Earth) has moved 6.9 billion km around the centre of the Milky Way Galaxy. Using referents outside the Milky Way Galaxy, it has, during the same time, moved some 55 billion km in space. So, I was born some 60 billion km away from wherever in space we actually are now. In the context of the Universe this is still local space, and I do not need to account for the very expansion of space. The looming collision of the Andromeda Galaxy speeding towards us is still 4.5 billion years away and irrelevant in the scale of my lifetime. Taking my present location as (0,0,0,0) and the X-axis as the straight line from where we were then to now, the coordinates of my birth location become (-60, 0, 0, -73 years) where x, y and z are measured in billions of km.

Everything is relative to here and now.

It is not even certain that time is actually a dimension of the same kind as the 3 physical dimensions we perceive. Some think it emerges together with the Universe and everything. Others think it is just a facet of existence. In any event, we only experience it as a backdrop for, but separate to, our own individual existences. On the stage of our existences, we strut and fret our hours away, but we cannot interact with or impact the relentlessly moving backdrop. Just as with nonsense rhymes using language, nonsense equations by learned physicists about the theoretical access to past times is just nonsense. As with any language, mathematics can also describe the unreal and the nonsensical. Speculative cosmologists have more in common with Edward Lear than they would like to think.

Considering time to be a dimension is no more than a convention or, at best, an analogy. It does not help either, that

  1. we have no clear definition (or understanding) of what a dimension is, and
  2. we can take a dimension to be anything that can be counted.

We can measure the oscillation of apparent motions and assume that such motion is regular and then infer the passage of time. But what time is other than a magical, necessary backdrop for everything is beyond our comprehension. We cannot be certain that a second now is the same, or longer, or shorter, than a second at some other time. (A second now must be longer than a second was then).

The world is what our perception tells us it is. But our perception is limited, and it limits the boundaries of our reality. We perceive space and everything around us as having 3 dimensions, yet we cannot truly conceive of any real thing having other than three spatial dimensions. In our 3-dimensional world we can define one- and two-dimensional things only as concepts (lines and surfaces) but we cannot identify any real-world objects which have only one or two dimensions. Moreover, real things having more than 3 dimensions are beyond our comprehension. How a fourth spatial dimension could be manifested lies outside of human reason. We have the language to describe – but only conceptually – any number of dimensions. Scientists and mathematicians speculate about 3 or 7 or 9 or infinite dimensions and claim either that 3 is the most probable or theorise that the others are hidden in the strings that make up the world, but the human brain can only perceive 3. (I note in passing that invoking the infinite is itself an admission of incomprehensibility). It is a fruitless and inevitably circular discussion to question whether it is our perception which is limited to 3 dimensions or whether the universe has only three to be perceived. Our universe is enabled, and strictly constrained, by what our cognition allows us to perceive. Every real thing in our universe has three spatial dimensions; no less, no more. Our universe has 3 spatial dimensions because that is all, and only what, we can perceive.

I probably read “Flatland” as a teenager where a sphere in Flatland can only be perceived as a circle.

Flatland: A Romance of Many Dimensions is a satirical novella by the English schoolmaster Edwin Abbott Abbott, first published in 1884 by Seeley & Co. of London. Written pseudonymously by “A Square”, the book used the fictional two-dimensional world of Flatland to comment on the hierarchy of Victorian culture, but the novella’s more enduring contribution is its examination of dimensions. – Wikipedia

No matter how many dimensions the universe may have, three dimensions is all human cognition can ever perceive. It is that reality which constrains all our thought. It becomes a fundamental assumption for science which the scientific method cannot penetrate. If other dimensions exist, then what we perceive in three are projections. As a shadow is perceived to be two-dimensional. But to have a projection or a shadow in our 3-dimensional world we would need some kind of cognitive light from the other, higher dimensions to create what we perceive.

But human cognition is limited. We cannot perceive what we cannot perceive. And we have no clue – earthly or divine – as to where and when we are.

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God or no-God? That is the wrong question

August 24, 2021

I find debates where one unprovable belief battles against other unprovable beliefs to be tiresome. Human cognition does not allow an absence of belief. A claimed non-belief is, of course, just another belief. I find statements of the kind “I do not know, but I know it isn’t that” to be self-contradictory and shallow. I find invocation of the scientific method, or of Divine Beings, without reference to boundary conditions and the limits of knowability, to be incomplete and invalid as arguments. This essay is just my attempt to marshal my own thoughts as to why I find it so.

(revised 26th August 2021)

The primal need to know

Human cognition demands that the world around us is ordered and rational and that it is capable of being understood. This is the fundamental and overriding assumption that pervades all thought and human endeavour. Science begins with this assumption of order in all parts of the universe and throughout all time. In addition, science assumes that causality, and the arrow of time, apply. The human brain is finite, and its attendant senses are limited. We do extend our sensory range with instruments but even these must convert their detected signals to be what humans can perceive directly by their senses and interpret with their brains. Even when looking at the same thing, what the eyes of a dog see as interpreted by a canine brain, is different to what a human eye sees as interpreted by a human brain. Whatever, and all, that humans observe are just their perceptions and are always subjective.

Human comprehension is “cabined, cribbed, confined, bound in” by the capabilities of the brain/sense combination. That some things are unknowable to the finite human mind is inevitable. Reason tells us that knowledge is whatever a brain can comprehend to be knowledge. Epistemology is a whole branch of philosophy dedicated to understanding the nature of knowledge. And yet we can do little better than concluding from all our analyses that knowledge is what knowledge is. Whatever lies beyond human comprehension is the realm of the unknowable. Observation is limited to what exists and is observable by the human brain/sense combination. Human reason and comprehension, though finite, can contemplate the unknowable, but without any hope of comprehension. Human language (including mathematics) can encompass both the real and the unreal. Language can describe things that do not exist. But when language addresses the unknowable it only deals in labels. The brain using that language cannot wrap itself around the unknowable which it labels. What we do not know, we seek and sometimes find. What we cannot know we can never find, and yet we seek. I cannot help but conclude that it is the seeking, not the acquisition, of knowledge, which is a primal characteristic (perhaps a purpose) of the human species.

But the ego of human cognition is such that it does not allow the inexplicable to remain dangling and unaddressed. That all mysteries must have, and require, explanation is a distinguishing characteristic of the human species. Human cognition abhors a mystery without an explanation. Anything mysterious or incomprehensible is always, eventually, explained away by an appropriate invocation of unknown, unknowable, imagined states or powers. It applies as much to science (metaphysics) as to philosophy or to theology. The explanations are often just labels for speculations which harness the unknown and the supernatural. Reason tells us that everything has a cause, but the First Cause problem defeats us. Our language allows us to formulate self-contradictory impossibilities. Before the beginning, we say and after the end. When our reason and our logic make recourse to infinite regress, it is often dismissed because it is incomprehensible. When we reach the unknowable, we invent new labels for abstract concepts such as divine or supernatural or infinity or forever. Never mind that they are unknowable. An infinite space or a forever existence are inherently incomprehensible and magical. We pretend that inventing a label brings us closer to understanding.

God Theories and the invention of Gods

In human discourse, the invention of God Theories and of Gods came about as we sought answers and explanations for inexplicable relationships observed in the surrounding physical world. The invocation of Sun-gods and wind-gods and weather-gods seems both obvious and inevitable in the early history of man. The invented explanations – again inevitably – required the existence of supernatural states and the exercise of powers beyond human capability. It was entirely logical and reasonable, then, that postulating the existence of supernatural or superhuman powers or states was entirely justified by the greater need to bring a perceived order to the observed world. To have a perceived order in the physical world was primal. The emotional human need for spirituality was also partly satisfied by the inclusion of the supernatural. With this world view, which allowed incomprehensible states and supernatural powers, these Great Explanations invoking the unknowable were quite reasonable, even if they were all merely labels for what could not be comprehended. Mysteries were replaced by labels which implied, but never actually bestowed, understanding. As human knowledge and sophistication increased, the need for the supernatural also adjusted to the new mysteries uncovered. It was not necessary for these supernatural states or powers to be invested in imaginary Beings, but some mediator was necessary for the exercise of supernatural powers. What the human mind considers reasonable is dynamic and shifts as learning occurs and knowledge increases. Given the existence of unknown, extra-natural forces, it was entirely reasonable, then, to imagine a living entity, a Being, as the mediator. Doing so certainly did improve the narrative. The first Sun God had the power to make the Sun rise every day. This explanation did not necessarily have to be invested in a Being, but it was convenient. It was merely for the ease of the narrative that these Beings took on forms and shapes and behaviours that were close enough to human forms and behaviours to be recognised and identified with. And so were born the Theories of Gods, the Gods, and the various pantheons of gods.

It is my contention that

  1. Human comprehension is finite and limited, and
  2. demands that order be perceived in the world.
  3. Incomprehensibilities and mysteries need to be explained to maintain the perceived order of the world, and
  4. invoking supernatural states and powers allows unanswerable questions and infinite regressions to be closed.
  5. Supernatural states and powers need a mediator, and therefore
  6. Beings invested with such states and powers have been invented as the mediators.

Inexplicabilities and mysterious events were most conveniently explained as the work of unknowable, supernatural or superhuman things. That these imagined things, labelled gods, were then imbued with the quality of being and of having forms and shapes and behaviour and families was, and is, mere embellishment. The fundamental reason for inventing any god was to be able to answer or explain the inexplicable. Every God ever invented was, at its core, a Theory of Explanation.

Religion, of course, is something else (see Notes).

The Great Mysteries

Where once we resorted to supernatural Beings, cosmology and metaphysics now resort to equally fanciful Theories. We have not had a new God invented for over a thousand years. We have had many Theories propounded though. The theories of today are consistent with the knowledge of today. They are as astute (or as ridiculous) as the God Theories of old.  The number of theories in cosmology about the origins of the universe are as numerous today as the multitude of ancient creator gods. The common feature is that these Theories, then as now, speculate about the incomprehensible and the unknowable. I observe that atheism today is very often all about debunking these God Beings, these deities in the image of man. They often focus on the attributes of the invented Beings. But this is superficial, and atheism rarely addresses the great mysteries which led to the invention of the God Theories in the first place.

What is considered inexplicable has changed over the years though many of the inexplicabilities have only changed cosmetically and in formulation. None of the great, deep mysteries about life the universe and everything have changed much or been resolved. The ultimate questions about the physical world regarding matter and energy and motion have not vanished either; they have just become much more sophisticated. We now say we know why the earth rotates around the Sun though we still have no clue as to how gravity is mediated. We know exactly how the effects of gravity can be calculated and we can traverse the distant reaches of space. But we have no clue as to why the perceived force follows an inverse square law and not the inverse cube law or something else. We can calculate the effect of electro-magnetic fields and can generate light and electricity and heat and motion almost at will. But we still have no clue as to how the earth communicates to a raindrop that it must move towards the surface of the earth. When we throw a ball in the air, we still have no idea of how the earth tells the ball it is time to change direction and fall. Einstein’s spacetime would say that the earth’s gravity distorts the fabric of the spacetime in which the ball exists such that it has no choice but to move down the spacetime slope towards the earth. But “down” is defined by gravity. To “move down” merely invokes another form of some kind of super-gravity, for there is no reason for any motion up or down a slope unless there is an overriding force. (When in doubt we can always just invent a new fundamental particle imbued with supernatural powers as the mediator. Let us call it a graviton). Nevertheless, we believe we now know why the earth orbits the Sun and have consigned the Sun-gods to the realm of the redundant. But the stark reality is that we still do not know how gravity is mediated.  It is noteworthy that the word gravitation could be replaced by the words magical attraction in any scientific text without any loss of meaning. (See notes). Admitting to magic, however, is not politically correct or acceptable. Instead, we now invoke the Theories of Relativity and Quantum Mechanics with spooky actions shrouded in mystery, except that we no longer label these metaphysical Theories as Gods.

Quantum mechanics dreams of a single all-encompassing quantum wave function, as just one particular instance of an incomprehensible infinity of possible wave functions. This Ultimate Wave Function which happens, by chance, to collapse to give all the other wave functions which in turn give us and the universe which we inhabit. One Ultimate Wave Function to rule them all. The quantum vacuum is not quite completely empty since it has the laws of quantum mechanics embedded within it. It is another Theory of Explanation which might as well be called the God of Quantum Mechanics. Though I am not clear if the God of Random Interaction is the Son or the Father. With the God of Gravity they make up a Trinity.

All the Great Mysteries – Existence, Causality, Time, the arrow of time, Life, Identity, Consciousness, Spirituality, Ethics, and Morality – have been great mysteries for at least 10,000 years and are still great mysteries. I do not include Mathematics in my list of Great Mysteries. I take the view that Mathematics is an invented language describing relationships in our observed world. Like all languages, it can also describe things that do not exist or are ridiculous. The beauty of relationships observed in the real world are not due to Mathematics, but due to the mystery of Existence and as described by Mathematics. Beauty lies in the thing not in the language describing the thing.

Mathematics started in prehistory with counting and the study of shapes

The metaphysics of existence remain mystical and mysterious and beyond human cognition, as much today as in prehistoric times. Nevertheless, it is the cognitive capability of having the concept of a unique identity which enables the concept of one. ……. Numbers are not physically observable. The concept of one does not, by itself, lead automatically to a number system. That needs in addition a logic system and invention (a creation of something new which presupposes a certain cognitive capacity). It is by definition, and not by logic or reason or inevitability, that two is defined as one more than the identity represented by one, and three is defined as one more than two, and so on. Note that without the concept of identity and the uniqueness of things setting a constraint, a three does not have to be separated from a two by the same separation as from two to one. The inherent logic is not itself invented but emerges from the concept of identity and uniqueness. That 1 + 1 = 2 is a definition not a discovery. It assumes that addition is possible. It is also significant that nothingness is a much wider (and more mysterious and mystical) concept than the number zero. Zero derives, not from nothingness, but from the assumption of subtraction and then of being defined as one less than one. That in turn generalises to zero being any thing less than itself. Negative numbers emerge by extending that definition. The properties of zero are conferred by convention and by definition. Numbers and number systems are thus a matter of “invention by definition”, but constrained by the inherent logic which emerges from the concept of identity.

We have discovered and can enumerate more Laws of Nature now than we ever could. But why these particular laws exist and none other, remains a Great Mystery. The existence of the natural laws, of matter, of energy and even of space itself remain as answers to unknown questions. We do not know what compels them to be what they are. The spark of life remains elusive. If nothing else the apparent purpose of all living things to survive, grow and replicate has appeared from Chance knows where. A purposeful chance is to delve into the unknowable. 

They all boil down, in my view, to two fundamental Great Mysteries. I find that they can be grouped either under Existence or under Life. Causality, Time, and Identity all emerge from the Great Mystery of Existence. It seems logical to me that Consciousness, Spirituality, Ethics, and Morality are mysteries which follow from the Great Mystery of Life. I used to subordinate the Life Mystery to the Mystery of Existence, but I think the injection of an apparent purpose elevates Life to be as great a mystery as Existence.

The fundamental questions

What leads to life the universe and everything?

God or no-God? is, in my view, a rather shallow question. There is no great mystery in the invention of gods. Every God ever invented was, at its core, a Theory of Explanation. And even if we get to find the God of the Theory of Everything, we would still have to reach for the unknowable to comprehend Existence and make sense of Life.

The fundamental Great Mysteries have always been, and still are:

  1. Why existence? and
  2. Why life?

Note 1.

Gods need to be distinguished from religions.

My take on religions is that they came later, after beliefs in gods had caught the human fancy. They came together with, or because of, an increasing need for human societies to organise themselves. They provided a way for the exercising of political power by utilising the human need for spirituality and exploiting the established beliefs in gods. All organised religions, whether they admit to it or not, are attempts to influence the behaviour of others and are all, unavoidably, cases of exploiting belief for the exercise of political power. I find the lip-service paid to the separation of state and religion rather meaningless since all organised religions – as all states and political parties – are involved in the business of influencing the behaviour of others.

Note 2.

Take, for example, this text from the Wikipedia article on Gravity where I have replaced the words “gravity” and “gravitation” with “magical attraction”.

Gravitation Magical Attraction, is a natural phenomenon by which all things with mass or energy—including planets, stars, galaxies, and even light —are attracted to (or gravitate toward) one another. On Earth, gravity magical attraction gives weight to physical objects, and the Moon’s gravity magical attraction causes the tides of the oceans. The gravitational magical attraction of the original gaseous matter present in the Universe caused it to begin coalescing and forming stars and caused the stars to group together into galaxies, so gravity magical attraction is responsible for many of the large-scale structures in the Universe. Gravity Magical Attraction has an infinite range, although its effects become weaker as objects get further away.

Gravity Magical Attraction is most accurately described by the general theory of relativity (proposed by Albert Einstein in 1915), which describes Gravity Magical Attraction not as a force, but as a consequence of masses moving along geodesic lines in a curved spacetime caused by the uneven distribution of mass. The most extreme example of this curvature of spacetime is a black hole, from which nothing—not even light—can escape once past the black hole’s event horizon. However, for most applications, Gravity Magical Attraction is well approximated by Newton’s law of universal gravitation Magical Attraction, which describes Gravity Magical Attraction as a force causing any two bodies to be attracted toward each other, with magnitude proportional to the product of their masses and inversely proportional to the square of the distance between them

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Why don’t dogs do philosophy?

June 30, 2021

Philosophy is a product of idle minds.

First and foremost, minds are concerned with survival and the basic necessities of staying alive. Breathing, escaping predators, food and shelter – in that order. Having experienced severe asthma attacks many years ago, I put breathing unequivocally first. When struggling for breath there is no other thing, no pain, no emotion, no other higher-order need which can, or does, occupy the mind. The entire focus of the brain and the body is solely on getting the next gulp of air. It is only after the basic physiological need are satisfactorily met that the mind has the time over to be idle. It is only then, and providing that the body is not so exhausted or under physical stress such that the mind does not embrace sleep, that the mind feels free to wander down non-essential – but interesting – paths. And that leads to philosophy.

The word philosophy originates from the Greek meaning for love of wisdom. In practice, it is the formulation of fundamental, unnecessary, unanswerable questions about life, existence, reality and everything. Having formulated the unanswerable question, philosophy then spends hours of thought, circular arguments, a mountain of verbiage, a flood of circumlocution and reams of paper in proposing unsatisfactory answers. The greater the number of paradoxes that can be introduced into an argument, the deeper and more profound is the philosophy. (Metaphysics is merely a particularly obscure – but not necessarily more obtuse – part of philosophy). There is no philosophical question which can not be put off till another time; which cannot be interrupted for more important survival tasks; which cannot be abandoned for a glass of wine. However, a glass or two can aid the production of even deeper, more incomprehensible answers to the unanswerable questions. There is no answer which needs an action to be taken. There is no philosophy which cannot be replaced by another. Philosophical questions may be about the fundamental nature of life but they are not questions which need answers for day-to-day living. There is no urgency or deadline for finding answers. A professional philosopher has no deliverables – or at least none which need to make sense. Philosophy is an occupation for minds which are free from the burden of resolving the basic physiological needs of the bodies they inhabit. Philosophy is an occupation for idle minds.

But this raises another philosophical question. (Unanswerable of course).

Domesticated dogs have found a sustainable solution for their basic physiological needs. They are protected from predatory threats, have food provided at regular intervals and are well sheltered from the elements. They are not used by their human master-slaves as food (except in Korea). They are generally well respected by their master-slaves in sickeness and in health. The solution does not come free. They have to accept the curtailment of many freedoms. Their human master-slaves control their movement, their diet, their friends and their breeding partners. Nevertheless, their minds are entirely and wonderfully free from the burdens of meeting their fundamental physiological needs. Their idle minds have all the time they could possibly need for unnecessary, unanswerable questions.

So why don’t dogs do philosophy?

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