“Mathematics, by the virtue of consisting of nothing other than positional names, preserves constant relations, since only constant relations are expressible in the grammar of mathematics: the grammar of positional names.”
“The discipline we call computer science is more generally the logic of operations, and is superior in informational density to mathematics in that it is causal and mathematics is merely descriptive.”
THE FOUNDATIONS OF EVERYTHING: DIFFERENCES
differences( logic ) > speech > sets > agorithms > mathematics
MATH CONFUSED US.
In mathematics, at least, for the most part, the means of conducting operations to solve a problem is nearly identical to the means of demonstrating the construction of a solution using existentially possible operations.
We sought to copy mathematics – starting with the Greeks. But we lacked the understanding of why math was so effective at the ascertaining truth of relations: because there is very little difference between the process of theorizing and the process of construction.
FOUNDATIONS OF LOGIC
The foundations of logic like those of mathematics are terribly simple as subsets of reality. But by doubling down in the 19th and 20th centuries all we have found is that we say rather nonsensical terms like ‘the axiom of choice’ or ‘limits’ rather than ‘undecidable without appeal to information provided by existential context’. After all, math is just the discipline of scale-independent measurement, and the deduction that is possible given the precision of constant relations using identical unitary measures. Logic is nothing more than than set operations. Algorithms are nothing more than sequential operations restoring time. Operations are nothing more than algorithms restoring physical transformation, time and cost. etc.
As a consequence, I find most of this kind of terminological discourse … silly hermeneutics. As Poincare stated ‘that isn’t math its philosophy’. Or as I would say, ‘with platonism we depart science and join theology. It may be secular theology in that it is ideal rather than supernatural, but it is theology none the less’.
it is one thing to say ‘by convention in math (or logic or whatever dimension we speak of) we use this colloquialism (half-truth) as a matter of convenience. It is not ‘true’. It is just the best approximation given the brevity we exercise in simplifying our work.
There exists only one possible ‘True’: the most parsimonious and correspondent testimony one can speak in the available language in the given context. Everything else is a convention.
Ergo, if you do not know the operational construction of the terms that you use, you do not know of what you speak. That does not mean you cannot speak truth any more than monkey cannot accidentally type one of the Sonnets.
This is why the operationalist movement in math we call Intuitionism failed.
Anyway. Well-formed (grammatically correct) statements in math may or may not be decidable but our intention is to produce decidable statements. In symbolic logic, well formed (grammatically correct) statements may or may not be decidable. in logic (language), well-formed (grammatically correct) statements are difficult to construct because of the categorical difference between constant relations (ideals in math), constant categories (ideals in formal logic), and inconstant categories (ordinary language). Furthermore, the process of DEDUCTION using premises (or logical summation) limits us to the utility of true statements. Ergo for that purpose statements can only evaluate to true or not-true (including false and undecidable). While for the purpose of INDUCTION (transfer of meaning by seeding free association, or the construction of possibility by the same means) seeks only possibility or impossibility not truth or falsehood.
How can you claim to make a truth proposition and demand precise language when your premises are mere demonstrably falsehoods used by convention?
“Nouns are names. Numbers are nouns. Numbers evolved as positional names. Numbers are positional names of constant relations. As positional names, they are extensions of ordinary language. Math: the science of measurement by the use of constant relations.”
We use many positional names: none, one, and some, short medium and tall; small, medium, and large; front, middle, and back; right center and left; port and starboard; daughter, mother, and grandmother;
Numbers differ from ordinary nouns only in that we produce them by positional naming. Whereas early positional names varied from one two and many, to base ten, or base twelve, or in the twenties, or sixties, each which increases the demand on the human mind; the decimal system of positional naming
Positional names are produced by a series of consistent operations. We call those series of consistent operations ‘functions’. By analogy we (unfortunately) called all such functions numbers: a convenient fiction.
Because of positional naming all positional names (numbers) are context independent, scale independent, constant relations, descriptively parsimonious and closed to interpretation.
So unlike other nouns (names), they are almost impossible to misinterpret by processes of conflation (adding information), and are impossible to further deflate (removing information).
Any other information we desire to add to the noun,( by which we mean name, positional name, number) must be provided by analogy to a context: application.
Numbers exist as positional names of constant relations. Those constant relations are scale-independent, context-dependent, informationally parsimonious, and nearly impossible to conflate with information that will allow for misinterpretation or deception.
As such, numbers allow us to perform DEDUCTIONS that other names, that lack constant relations, scale independence, context dependence, parsimony, immutability, and incorruptibility do not. Because deduction is possible wherever constant relations, parsimony, immutability, and incorruptibility are present.
As such, numbers serve as a method of verbal reasoning within and beyond the limits of human imagination (cognition), short term memory, and ordinary reason.
Numbers then are simply a very clean set of nouns(positional names), verbs (operations and functions), including tests of positional relations (comparison operators) that allow us to describe, reason and discourse about that which is otherwise beyond our ordinary language, and mental capacity.
As such we distinguish language, reason, and logic from numbers and measurement, and deduction both artificially and practically. Since while they consist of the same processes, the language of numbers, measurements, and deductions is simply more precise than the language of ordinary language, reason, and logic, if for no other reason than it is nearly closed to ignorance, error, bias, wishful thinking, suggestion, obscurantism, deceit, and the fictionalism of superstition, pseudo-rationalism, pseudoscience.
Unfortunately, since to humans, that which allows them to perform such ‘seeming miracles’ that are otherwise beyond comprehension, must be justified, we invented various fictionalisms – primarily idealisms, or what philosophers refer to as platonisms – (mythologies) to explain our actions. To attribute comprehension to that which we did not comprehend. To provide authority by general rule to that which we could only demonstrate through repeated application. So mathematics maintains much of it’s ‘magical language’ and philosophers persist this magical language under the pseudo-rational label of ‘idealism’ or ‘abstraction’. Which roughly translates to “I don’t understand”.
Perhaps more unfortunately, in the 19th century, with the addition of statistics and the application of mathematics to the inconstant relations of heuristic systems: particularly probability, fiat money, economics, finance, banking and commercial and tax accounting, this language no longer retains informational parsimony, and deducibility, and has instead evolved into a pseudoscience under which ignorance, error, bias, wishful thinking, suggestion, obscurantism, and deceit are pervasive.
Math is a very simple thing. It’s just ordinary language with positional names that allow us to give names and describe transformations to, that which is otherwise beyond our ability to imagine and recall, and therefore describe or reason with.
Like everything else, if you make up stories of gods, demons, ghosts and monsters, or ‘abstractions’ or ‘ideals’ you can obscure the very simple causality that we seek to discover through science: the systematic attempt to remove error, bias, wishful thinking, suggestion, obscurantism, fictionalism, and deceit from our language of testimony about the world we perceive, cognate, remember, hypothesize within, act, advocate, negotiate, and cooperate within.
Numbers are positional names of context-independent, scale-independent, informationally parsimonious, constant relations and mathematics consists of the grammar of that language.
In other words, Math is an extension of ordinary language, ordinary reason, and ordinary science: the attempt by which we attempt to obtain information about our world within, above, and below human scale, by the use of rational and physical instrumentation, to eliminate ignorance, error, bias, and deceit from our descriptions, and as a consequence our language, and as a consequence our collective knowledge.
MATH IS SIMPLE
The foundations of mathematics are simple.
The fact that they even phrase the question as such is hysterical. The reason mathematics is so powerful a tool is precisely because its foundations are so trivial. Like discourse on property in ethics and law, it is a word game because no one establishes sufficient limits under which the general term obscures a change in state.
Math is very simple. Correspondence (what remains and what does not), Types, operations, grammar, syntax. Generally we use mathematics for the purpose of scale independence. in other words, we remove the property of scale from the set of correspondences. But we might also pass from physical dimensions to logical dimensions (there are only so many possible physical dimensions). So now we leave dimensional correspondence. In mathematics we remove time correspondence by default, and only add it in when we specifically want to make use of it. In sets we remove temporal and causal correspondence … at least in most cases. So we can add and remove many different correspondences, and work only with reciprocal (self referencing) correspondence (constant relations). But there is nothing magic here at all except for the fields (results) that can be produced by these different definitions as we use them to describe the consequences of using different values in different orders.
But if you say “I want to study the parsimony, limits, and full accounting, of this set of types using this set of operations, with the common grammar and syntax” that is pretty much what someone means when they say ‘foundations’. Most of the time. Sometimes they have no clue.
There is nothing much more difficult here in the ‘foundations’ so to speak. What’s hard in mathematics is holding operations, grammar and syntax constant, what happens as we use different correspondences (dimensions), types, and values in combination with others and yet others, to produce these various kinds of patterns that represent phenomenon that we want to describe. And what mathematicians find beautiful is that there is a bizarre set of regularities (that they call symmetries or some variation thereof), that emerge once you becomes skilled in these models, just like some games become predictable if you see a certain pattern.
But really, math is interesting because by describing regular patterns that produce complex phenomenon, we are able to describe things very accurately that we cannot ‘see’ without math to help us find it.
Its seems mystical. It isn’t. Its just the adult version of mommy saying ‘boo’ to the toddler and the joy he gets from the stimulation. There is nothing magical here. it’s creative, and interesting, but it’s just engineering with cheaper tools at lower risk: paper, pencil, and time.
Simplicity is necessary in mathematics since mathematical symbols and operations itself (state and operators) are necessary to allow us to remember state with sufficient precision that we can conduct comparisons between states.
However, if we restated the foundations of mathematics operationally (constructively – analogous to gears), and we stated the foundations of mathematical deduction negatively, as geometry, we would be able to show that it is convergence between the via-positiva construction, and the via-negative deduction that leads us to truth.
Unfortunately, man discovered (logically so) geometry prior to gears, and as such, we retain the ‘superstitious’ language of geometry (and algebra) of the superstitious era in which both were invented.
Reality has only so many dimensions. By adding and removing dimensions from consideration we simplify the problem of describing the constant relations within it.
Mathematics specializes in the removal of (a) scale, and (b) time, and (c) operations (and arguable (d) morality) from consideration, leaving only identity, quantity, and ratio, to which we add positional naming (numbers). We then construct general rules of arbitrary precision (scale independence) and apply those to reality wherein we must ‘hydrate’ (reconstitute) scale, time, and operations(actions).
So just as philosophy is ‘stuck’ in non contradiction instead of increasing dimensions in order to test theories, mathematics is ‘stuck’ in non-contradiction instead of re-hydrating (restoring dimensions) to justify propositions.
In other words, fancy words like ‘limits’ or ‘non-contradictory’ or ‘axiom of choice’ and various other terms in the field are just nonsense words that prevent the conversion of mathematics from a fictionalism into a science.
UNREASONABLE EFFECTIVENESS? NONSENSE.
The “Unreasonable effectiveness” trope annoys the hell out of me. The only reason this ‘magical mathematics’ nonsense perpetuates, and the average person is still afraid of mathematics, is because it’s taught as a superstition.
Math is trivial. 1 = any unitary measure. By the combination of some number of symbols – in the current case 0123456789, we can create positional names. By adding, subtracting units, and by adding and subtracting sets of units (multiplication and division), we can create positional names (numbers) for an unlimited set of positions. we can create names of positions in an unlimited number of directions (dimensions). We can create positions relative to any other position (relative positions). We can create changes in positions of relative positions. producing numbers, sets, and fields, and topographies (many different fields.
So the fact that math is ‘unreasonable’ is rather ridiculous. It’s people who are unreasonable. Math is TRIVIAL. Deduction in multiple dimensions is hard because we are not well suited to it.
I mean, we have 26 letters, and 44 phonemes in the english language. If we were ‘elegant’ we might increase the 26 to 44 letters, so that english was easier to read. but look at what we can say with those 44 phonemes, 26 characters, and 250K words in some including terms, and maybe 200K words that are not archaic.
There are roughly 100,000 word-families in the English language.
A native English speaking person knows between 10,000 (uneducated) to 20,000 (educated) word families.
A person needs to know 8,000-9,000 word families to enjoy reading a book.
A person with a vocabulary size of 2,500 passive word-families and 2,000 active word-families can speak a language fluently.
Of those we can pretty much COMMUNICATE anything, although in wordy prose, with only 300 words.
Now think of how much MORE you can say in language than you can say in mathematics.
Why should it surprise you that running around with a perfectly scalable yardstick that can measure any distance, allows you to measure and compare anything? It shouldn’t. It’s freaking obvious.
REMOVING MATHEMATICS FROM PHILOSOPHY AND THEOLOGY AND RETURNING IT TO THE SCIENCE OF MEASUREMENT.
In mathematics, construction must be operationally possible (computable), even if the descriptions (proofs) are only deducible.
Others only provide an IDEAL (logical) justification of why cantor is wrong, and not a REAL (scientific and operational) explanation of why he was wrong: that the technique (like gears) demonstrated something valuable: that the rate of production of positional names produces different sized sets regardless of the point of termination (scale or limit). Cantor is one step removed from theology(ideal by design), and speaking in philosophy (ideals), instead of speaking in mathematics (measurement) and science (operations).
The depth of this statement allows us to repair mathematics and return it to a science of measurement, rather than this nonsensical platonism used today – a remnant of the ancient greeks.
—“You’re saying all mathematical statements are true or false but the liar paradox is one example of an ordinary language sentence which hasn’t got a truth-value, right? Well, stated that way, I’d say you’re right about all of that, but are you also saying that the liar sentence expresses a proposition? That might be the part where it starts to get problematic.”—
In short, we can ask a question, or we can assert an opinion, conflate the two, or we can speak nonsense. And only humans (so far) can ask, assert, conflate, and fail at all of them. But out of convenience, we subtract from the real to produce the ideal, and speak of the speech as if it can act on its own.
Just to illustrate that the test we are performing (context) limits both what we are saying and what we can say. From the most decidable to the least:
1 – The mathematical category of statements, (tautological) single category. (relative measure)
2 – The ideal category of statements, (logical) multiple categories. (relative meaning)
3 – The operational category of statements (existential possibility)
(sequential possibility )
4 – The correspondent (empirical) category of statements. all categories. ( full correspondence )
5 – The rational category of statements ( an actor making rational choices) (‘praxeological’)
6 – The ‘moral’ category of statements ( test of reciprocity)
7 – The fully accounted category of statements (tests of scope)
8 – The valued (loaded) category of statements. (full correspondence and loaded with subjective value)
9 – The deceptive category of statements (suggestion, obscurantism, fictionalism, and outright lying.
We can speak a statement in any one or more of these (cumulative) contexts.
So for example, statements are not true or false or unknowable, but the people who speak them speak truthfully, falsely, or undecidedly. So performatively (as you have mentioned) only people can make statements.
However, to make our lives easier, we eliminate unnecessary dimensions of existence unused in our scope of inquiry, and we conflate terms across those dimensions of existence, and we very often don’t even understand ourselves what we are saying. (ie; a number consists of a function for producing a positional name, from an ordered series of symbols in some set of dimensions. Or, only people can act and therefore only people can assert, and therefore no assertions are true or false, the person speaking speaks truth or falsehood. etc.)
This matters primarily because no dimensional subset in logic closed without appeal to the consequence dimensional subset. In other words, only reality provides full means of decidability.
Or translated differently, there just as there is little action value in game theory and little action value in more than single regression analysis, there is little value after first-order logic, since decidability is provided by appeal to additional information in additional dimensions rather than its own. Which is, as far as I know, the principal lesson of analytic philosophy and the study of logic, of the 20th century.
Or as I might restate it, we regress into deeper idealism through methodological specialization than is empirically demonstrable in the value returned. Then we export these ‘ideals’ as pseudosciences to the rest of the population. This leading to wonderful consequences like the Copenhagen consensus. Or the many-worlds hypothesis, or String Theory. Or Keynesian economics. Or the (exceedingly frustrating) nonsense the public seems to fascinate over as a substitute for numerology, astrology, magic, and the rigorous hard work required
THE STATE OF MATHEMATICAL ECONOMICS
Understanding advanced mathematics of economics and physics for ordinary people.
The Mengerian revolution, which we call the Marginalist revolution, occurred when the people of the period applied calculus ( the mathematics of “relative motion”) to what had been largely a combination of accounting and algebra.
20th century economics can be seen largely as an attempt to apply the mathematics of relative motion (constant change) from mathematics of constant categories that we use in perfectly constant axiomatic systems, and the relatively constant mathematics of physical systems, to the mathematics of inconstant categories that we find in economics – because things on the market have a multitude of subsequent yet interdependent uses that are determined by ever-changing preferences, demands, availability, and shocks.
Physics is a much harder problem than axiomatic mathematics. Economics is a much harder problem than mathematical physics, and before we head down this road (which I have been thinking about a long time) Sentience (the next dimension of complexity) is a much harder problem than economics.
And there have been questions in the 20th century whether mathematics, as we understand it, can solve the hard problem of economics. But this is, as usual, a problem of misunderstanding the very simple nature of mathematics as the study of constant relations. Most human use of mathematics consists of the study of trivial constant relations such as quantities of objects, physical measurements.
Or changes in state over time. Or relative motion in time. And this constitutes the four dimensions we can conceive of when discussing real-world physical phenomenon. So in our simplistic view of mathematics, we think in terms of small numbers of causal relations. But, it does not reflect the number of POSSIBLE causal relations. In other words, we change from the position of observing a change in state by things humans can observe and act upon, to a causal density higher than humans can observe and act upon, to a causal density such that every act of measurement distorts what humans can observe and act upon, by distorting the causality.
One of our discoveries in mathematical physics, is that as things move along a trajectory, they are affected by high causal density, and change through many different states during that time period. Such that causal density is so high that it is very hard to reduce change in state of many dimensions of constant relations to a trivial value: meaning a measurement or state that we can predict. Instead we fine a range of output constant relations, which we call probabilistic. So that instead of a say, a point as a measurement, we fined a line, or a triangle, or a multi dimensional geometry that the resulting state will fit within.
However, we can, with some work identify what we might call sums or aggregates (which are simple sets of relationships) but what higher mathematicians refer to as patterns, ‘symmetries’ or ‘geometries’. And these patterns refer to a set of constant relations in ‘space’ (on a coordinate system of sorts) that seem to emerge regardless of differences in the causes that produce them.
These patterns, symmetries, or geometries reflect a set of constant relationships that are the product of inconstant causal operations. And when you refer to a ‘number’, a pattern, a symmetry, or a geometry, or what is called a non-euclidian geometry, we are merely talking about the number of dimensions of constant relations we are talking about, and using ‘space’ as the analogy that the human mind is able to grasp.
Unfortunately, mathematics has not ‘reformed’ itself into operational language as have the physical sciences – and remains like the social sciences and philosophy a bastion of archaic language. But we can reduce this archaic language into meaningful operational terms as nothing more than sets of constant relations between measurements, consisting of a dimension per measurement, which we represent as a field (flat), euclidian geometry (possible geometry), or post Euclidian geometry (physically impossible but logically useful) geometry of constant relations.
And more importantly, once we can identify these patterns, symmetries, or geometries that arise from complex causal density consisting of seemingly unrelated causal operations, we have found a constant by which to measure that which is causally dense but consequentially constant.
So think of the current need for reform in economics to refer to and require a transition from the measurement of numeric (trivial) values, to the analysis of (non-trivial) consequent geometries.
These constant states (geometries) constitute the aggregate operations in economies. The unintended but constant consequences of causally dense actions.
Think of it like using fingers to make a shadow puppet. If you put a lot of people together between the light and the shadow, you can form the same pattern in the shadow despite very different combinations of fingers, hands, and arms. But because of the limits of the human anatomy, there are certain patterns more likely to emerge than others.
Now imagine we do that in three dimensions. Now (if you can) four, and so on. At some point we can’t imagine these things. Because we have moved beyond what is possible to that which is only analogous to the possible: a set of constant relations in multiple dimensions.
So economics then can evolve from the study of inputs and outputs without intermediary state which allows prediction, to the study of the consequence of inputs and the range of possible outputs that will likely produce predictability.
in other words, it is possible to define constant relations in economics.
And of course it is possible to define constant relations in sentience.
The same is true for the operations possible by mankind. There are many possible, but there are only so many that produce a condition of natural law: reciprocity.
Like I’ve said. Math isn’t complicated if you understand that it’s nothing more than saying “this stone represents one of our sheep”. And in doing so produce a constant relation. all we do is increase the quantity of constant relations we must measure. And from them deduce what we do not know, but is necessary because of those constant relations.
Math is simple. That’s why it works for just about everything: we can define a correspondence with anything.
As far as I know, all truth refers to testimony and we use the term ‘True’ ‘loosely’ for many purposes – largely ‘consent’. Technically speaking logic gates output charges (1) or not (0).
We equate this to True=On (constant relation) or false=Off (inconstant). We do this to conflate the logically true (constant relations) and logically false (inconstant relations).
We do this DESPITE the fact that all logic is ternary with negative priority (1-False, 2-True, 3-Undecidable), because all premises are contingent. Since all premises are contingent, we cannot claim positives (constructions) are true, only that they are not false.
As a consequence we falsify alternatives leaving truth candidates as possibilities. This is in fact how cognition, communication, testimony, and science function: free association(some relations), hypothesis (meaning), theory(self-tested), “Law”(Market Tested). The only question is how we falsify.
In mathematics, logic, and language not all ideas can be constructed, and must be deduced by creating constructions that permit us to deduce that which we cannot construct (a heptagon being the most rudimentary problem in geometry – it cannot be constructed by ruler and compass).
Nearly all non-trivial constructions cannot be constructed (proven or testified to) they can only be described by the process of elimination.
Mathematics is an extremely simple logic since it consists of only one dimension: position. Models are constructed of just that one relation – but in large numbers. Language consists of many kinds of measurements. And is far harder to test. What we intuit as constant relations may be in our brains, but not in reality.
This isn’t something that’s open to opinion. Words consists of constant relations. There is simply much higher density that simple reductio models in more primitive grammars (logics).
GAIN: OPERATIONS (REAL) VS SETS (IDEAL) – CANTOR AS AN EXAMPLE OF THE PROBLEM IN MATHEMATICS AND BY EXTENSION EVERYTHING.
—”Ok but Cantor’s work is specifically set-theoretic, not analytical. Also, an infinite sum is by definition a sum over a countable set. So cantor’s notions are in fact relevant for this.”—Alex Pareto
Yes it is a sacred cow because people who are (knowingly or unknowingly) mathematical platonists are just as indoctrinated into superstitious nonsense as people who are indoctrinated into platonism proper, and people indoctrinated into theology. They know how to DO what they do (meaning make arguments with the objects, relations, and values of their vocabulary and grammar) but they don’t know how and why what they do functions.
Frequencies are the scientific description and infinities (sizes) the fictional (imaginary) description. The difference is that those of us who work in the sciences, where we CANNOT engage in Platonism, because that is the purpose of science: to prevent such ‘magical’ speech, and instead force us to undrestand the causal relations between reality and our speech.
So in this case a number consists of nothing more than the name of a position. That’s it. Mathematics consists of the vocabulary and grammar of positional names. Nothing more. Period.
We generate positional names by the process of positional naming. We can scientifically describe that process as did Babbage, Turing, and Computer Science (consisting of nothing but addition), with gears, or the positional equivalent of gears (positional names), or the electronic-switch(memory) of positional names, and use these gears to produce positional names and operations on positional names at varying speeds. We can also tell a ‘story’ about those things (a fiction) which is what we do with literary, symbolic, and set mathematics. And then we can tell a fairy tale about sets, as if they are an equivalent to red riding hood.
But no matter what we do, operationally, (scientifically) all we can do is produce a series of positional names faster or slower than another series of positional names.
Ergo, there exists only one name “infinity” for “unknown limit of operations” and different rates (frequencies) by which we generate positional names, using any set of operations with which we produce positional names.
This is why mathematics ‘went off the rails’ into fictionalism despite Poincare’s and others efforts at the beginning of the 20th century. Math is just the use of positional names which have only one property: position, and therefore only ONE constant relation: position.
All logic consists of the study of constant relations, and as such mathematics provides the most commensurable language of constant relations, since it has only ONE constant relation: position.