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
There is no generally accepted definition of what mathematics is. In the broadest sense it is a discipline concerned with the study of “quantity, structure, space, and change (arithmetic, algebra, geometry, and analysis)”. Though some claim that a number system is not an absolute necessity for mathematics, I do not think this holds in practice. I note that even the study of structure, space and change always assumes the ability to quantify. The application of any branch of mathematics requires quantification. (Which is why pure mathematics is considered, by pure mathematicians, to be morally superior and much more hygienic and refined than applied mathematics). It is suggested that the Greeks had geometry before arithmetic and that shapes and structures and relationships in our universe can be studied without numbers. Indeed they can, but this misses the point. It is the ability to count which is needed before geometry makes sense. To be able to say that one circle is an unknown amount bigger than another, or that triangles are similar but unquantifiably different is of limited value. Humans (and the Greeks) had a number system in place long before they came to geometry. Counting and numbers come first, before operators and strict rules of operation (a stringent grammar) allow the invention of arithmetic. The origin of counting by humans lies in pre-history, probably even before the 40 -50,000 year old tally-sticks that have survived. Numbers and shapes are not physical, material things. They are concepts and exist in the world of the immaterial. The history of mathematics is of an increasing level of complexity in abstractions which started with numbers and counting. Without a number system, mathematics reduces to descriptive text, which is available in any language. The ability to quantify, and therefore its manifestation as a number system, is, I think, integral to, and a core necessity for, all mathematics.
Mathematics does not really qualify as a science since no empirical observations are involved. It does conjecture and hypothesise. It does then test and prove, not by experiment, but by logic and reasoning. It does reach logical conclusions, both by induction and deduction. This logic in mathematics is the same logic which is empirically observed within the universe, together with rules of operation defined by the discipline. But mathematics is always, and only, about abstractions in the immaterial plane. If not science, is mathematics art? There is certainly a strong aesthetic quality to all branches of mathematics. It was my professor of pure mathematics, while doing an engineering course 50 years ago, who first instilled in me an appreciation for elegance as being a desired characteristic of good mathematics. Elegance, precision, simplicity, and conciseness are all considered qualities of good mathematics. I take art to be the human creation of unique artefacts for the emotional appreciation of that artefact by other humans. There is little doubt that such works of art are invented – not discovered. Certainly some indulge in mathematics for the sake of mathematics, for the aesthetic quality of the arguments and the reasoning and the conclusions. Pure mathematics comes very close to being art. But mathematics is not only art. It is a tool not only for scientists in their task of discovery, but also a tool used by virtually every field of human endeavour. It is primarily a tool for living. Engineers and lawyers and doctors and economists and traders all use mathematics in some form. Perhaps for pure mathematicians the emotions involved are similar to those of artists.
There is a point of view that mathematics lives within the very firmament and transcends all else. There is also the view that the beauty of the mathematical expressions and equations describing the relationships we perceive in the universe is the essence of spirituality. In a more extreme view, mathematics is considered the cause of existence. At one end of the spectrum of such views lies the quantum-mechanical view where everything derives from wave functions. Of course, every wave function is preceded by other, causal wave functions until there is just one wave function which rules them all. Then everything – existence itself, causality, time, logic and matter and anti-matter and energy – emerges as this one fortuitous, god-like, by-chance, Ultimate Wave Function collapses. The First Cause problem is conveniently bypassed by invoking the quality of cause-less, by-chance (random) origins for the UWF. The God of Random Things rules! OK!
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.
But I note that the logic that mathematics utilises is not its own but is the logic already prevalent in the universe it seeks to describe. I also note that it is existence and the Concept of One, of a singular, unique identity, which lies as the most basic assumption needed for any number system. Numbers are not real. They are adjectives much more than nouns. Every number system is existentially dependent upon the Concept of One. Then, by the application of a pre-existing logic, the set of a number system is built up by repeatedly adding One to itself. Addition is a logical step we can observe empirically around us, but which the number system needs to assume. The entire set of numbers exists because we discover that existence provides the concept of identity. That 3 is as distant from 2 as 2 is from 1 is an assumption. It is part of the grammar of the language which is numbers. All the wonderful “discoveries” of number theory are there to be discovered only because of the Concept of One and because we have defined the other numbers to be what they are. A circle is an abstraction. It is a two-dimensional concept we use in the three-dimensional world we perceive. The relationship between the concept of a circle and the concept of its diameter exists because things exist, not because mathematics exists. The value of pi (π) follows inevitably from existence not from mathematics. It takes the value it does as a consequence of our definition of numbers based on the Concept of One. Natural logarithms exist and e takes the value it does, not because things exist but because we define numbers as we do. We “discover” these only because we first invented numbers. The entire field of Number Theory still exists today because we are still “discovering” unknown details about the number system humans first invented in pre-history.
Numbers emerge from the concept of identity
Numbers are abstract. They do not have any physical existence. That much, at least, is fairly obvious and uncontroversial.
Are numbers even real? The concept of numbers is real but reason flounders when considering the reality of any particular number. All “rational” numbers (positive or negative) are considered “real numbers”. But in this usage, “real” is a label not an adjective. “Rational” and “irrational” are also labels when attached to the word number and are not adjectives describing the abstractions involved. The phrase “imaginary numbers” is not a comment about reality. “Imaginary” is again a label for a particular class of the concept that is numbers. Linguistically we use the words for numbers both as nouns and as adjectives. When used as a noun, meaning is imparted to the word only because of an attached context – implied or explicit. “A ten” has no meaning unless the context tells us it is a “ten of something” or as a “count of some things” or as a “measurement in some units” or a “position on some scale”. As nouns, numbers are not very pliable nouns; they cannot be modified by adjectives. There is a mathematical abstraction for “three” but there is no conceptual, mathematical difference between a “fat three” and a “hungry three”. They are not very good as adjectives either. “Three apples” says nothing about the apple. “60” minutes or “3,600” seconds do not describe the minutes or the seconds.
The number of apples on a tree or the number of atoms in the universe are not dependent upon the observer. But number is dependent upon a brain in which the concept of number has some meaning. All of number theory, and therefore all of mathematics, builds on the concept and the definition of one. And one depends, existentially, on the concept of identity.
It would seem that identity and the Concept of One emerge from existence and are embedded in our universe. But whether they do or not they are concepts we discover and which we try to describe using invented languages.
Croutons in the soup of existence
Numbers are abstract and do not exist in the physical world. They are objects (“words”) within the invented language of mathematics to help us describe the physical world. They enable counting and measuring. The logical one or the philosophical one or the mathematical one all emerge from existence and identity. Neither logic nor philosophy nor mathematics can explain what one is, except that it is. Every explanation or definition attempted ends up being circular. It is what it is. Mathematics presupposes that one exists but can only assume what it is.
The properties of one are prescribed by the assumptions (the “grammar”) of the language. One (1, unity), by this “grammar” of mathematics is the first non-zero natural number. It is the integer which follows zero. It precedes the number two by the same “mathematical distance” by which it follows zero. It is the “purest” number. Any number multiplied by one or divided by one remains that number. It is its own factorial. It is its own square or square root; cube or cube root; ad infinitum. One is enabled by existence and identity but thereafter its properties are defined, not discovered.
Mathematics is a language
Mathematics is a tool for the process of discovery which is science. It has a large aesthetic content. But first and foremost, it is a language to describe the universe around us and the relationships and patterns we discern. The pure mathematician practices his art, if art it is, using language of mathematics to express his discovered thoughts. He discovers wonders expressed in the invented symbology and grammar of mathematics. It is a language that allows the expression of abstract concepts not possible (or very difficult) with conventional languages. It provides precision and conciseness for describing increasingly complex, abstract concepts. The symbols and its grammar are an invented construct. There are discoveries in mathematics, but these are always discoveries about mathematics itself and are the unknown consequences of the assumptions used to invent the mathematics in the first place. Number theory discovers consequences due to the assumptions made to create the number system. Mathematics discovers detailed things about itself only after it has first been invented as a construct.
Mathematics is a language like all other invented language, albeit with a much more stringent and rigorous grammar than conventional language. It describes the relationships and patterns we find in the world around us, but it is not itself the relationships or the patterns. Not all mathematical sentences have meaning. Like all languages it provides a much wider capability of expression than is needed. Like all languages unforeseen rhymes are discovered after words are invented. Like all languages mathematics can produce incomprehensible text. And it can produce nonsense text.
“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”.
Mathematics is invented, not discovered.
Tags: Concept of One
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