Section 50 (last updated 02.18.2021)
Point in space is same as moment in time
Time being 1 dimensional means it is infinite. This is mathematically true as the number 1 is implied by every other number no matter the quantity. This is the view that numbers are qualities not just mere quantities, in other words there is a logical component to numbers being in their particular place and position in relation to other number. In ordinal number we import a logical order to numbers. That the number 1 is first in the order of all numbers after 0 solidifies the quality of being a single and distinct quantity no matter how many variables is grouped by it, the number 1 is the quality of an identity belonging to the same distinct group. This is the same with the quality of time being one dimensional, time is the sequence at every distinguishable step of an activity having the same identity. Jack is the single identity that groups together an infinite set of possible behaviours and situations, this infinity is disclosed within jack as their particular, or particle, acting on them.
Space is the realm where a single possible event is picked out at a single time over the other possible events. If I am standing close to the ledge of an oncoming train, it is a possibility for me to jump down into the tracks and be mangled by the train, this possibilities exists at one point in space where the tracks are, that point in space is where that possible event exists. but my self preservation instinct determined the event where I am off the ledge. Where you are in space determines on a basic level what possible event will turn real. This is because every point in space is occupied by a possible event, where you are spatially turns a possible event into a real one, which so far is possible is actually there occupying that space, and once it is met with the particular determination of it, actually going there, it becomes a real not just a possible event. But it is not simply the space that initiates the event but rather the space facilities a particular kind of relation between two things, the space where the track is facilities the kind of relation a man will have with an oncoming train.
The wavelength state exists as spectrum of possibilities, and the particle state, which determines this spectrum, or rather where this spectrum is contained within, chooses to visit any point on the wavelength to realize any possible event, what possible event it determines into reality will govern the direction of the length. If I choose to jump in the tracks, the length of that particle state stops there.
There are two different dynamic conceptions that makes an external and internal view point of the same phenomenon of time. The external view of time is that you see a set of objects in motion, that if one object is at one place it is not at another place at the same time, it has to move to that other place to occupy it which then leaves the place it was before empty or filled with something else. This is the law of non contradiction in logic and in elementary physics describes ordinary classical motion. The problem is that it does not explain a very simple problem in motion that the Zeno paradox hints at, which is how is the object able to even move to a different position in the first place?
This means how is the object able to maintain an identity throughout the change of position, and how is the change of position acquired if it was not part of the objects identity? Aristotle resolves the Zeno paradox by suggesting that something simply gets up and moves, or rather it has the self capacity to determine itself into another state. It is certainly true that things have a self determination, which is what Aristotle points out that there is a self determination, if not in the thing itself, there is at least a general self determination. But the problem with self determination is that whenever it determines itself in one way it is no longer the same as it was before, or if it is, there is now a difference between an other, so the state different than its identity must somehow be explained as part of its identity.
(How everything and each thing is a 1) (quantized system)
And the process between objects in varying dimensions seems to be an ascension akin to asking the mathematical question of derivative “an expression representing the rate of change of a function with respect to an independent variable.” The question of a derivative asks how can the number 1000000 be derived from the number 1? The answer that each real number is a seepage component on its own cannot answer this question because the addition of numbers forward suggest that the greater multiplicity of numbers are derived from the lesser, that we get the number 5 from 0.
The ancients answer the paradox of the derivative by claiming that division, or how the greater sum of things arrive from a less, is due because they are “relations”, that numbers are not entities but are relations. Looked at in this manners, and all seems to make sense, that the same quality, which is number, the form of all mathematical functions, is undergoing a self relation, and it is the complexity of the relations that is measurable as components. 2 is the self relation between 1 + 1.
Every number is a derivative of 1 by being a single number
when looking at a painting we do not necessarily notice its frame as anything distinct from its content but presuppose it as part holding the content of the painting which is what really should be focused on. However the frame of the painting for example has the function of separating the content of the artwork from the wall behind it, and there is its moment wherein the painting changes to the wall. The faculties or sense awareness has evolved to adopt to the rapid change of nature not merely in the way thing move around within a certain frame of reference, but rather the mind itself has evolved to get used to seeing the frame of reference itself, which is always constantly changeling, as stable and static. But in fact the very moment itself that discloses the operations of ordinary nature is constantly shifting and changing into a future event from a previous.
Now if this presupposes is a determination of what is already potentially true, and what is potentially true exists as a potential, in what sense is determination a freedom and not a mere crude determinism. In other words where is the freedom in a process that is simply fulfilling what it already knows to be true? For example, it is known that pushing a ball will move it from one place to another even without actually pushing it, therefore pushing it is redundant if the result is already known as moving from one place to another. The problem is that for the ball to move from one place to another presupposes that there is the force of pushing it, and once a force is presupposed, it cannot simply be known but is simultaneously also an action. Is the knowledge that pushing the ball moves it from one place to another prior to actually pushing it and then knowing that it will move?
Even if we say so, the quality of knowing itself then becomes the first action. Therefore the act to actually push the ball is done for the confirmation and demonstration of the already known fact that it will move from one place to another. And this is an aspect of freedom in the sense of determination because the fact that the ball moves from one place to another already presupposes the action of pushing. it does not really need to be confirmed, but because the choice was to go ahead and to confirm it anyways is made by actually pushing the ball and therefore constituting the knowledge that the ball moves from here to there is made, the determination to actualize an already preconceived fact is a property of freedom, because it does not need to have had happen, it was chosen to happen, to confirm itself, because it is itself anyways.
The infinite is the simultaneous occurring of all possible events form the view of a particular one in the bunch. A particular set of events having a sequence in time we know as a lifetime is a particular obscuration of the infinite into a focus on a finite event excluding all other events as simultaneous. This poses a problem in philosophy in the notion of reality. Do we call reality the particular experience of the individual, “my reality”, or is the reality the simultaneity of all possible events? The fact is that one does not exclude the other, one is only a limit of the other, but that limo itself is a delusion as other factors are excluded for the sake of the conception of one. But at the same time this so called delusion is a function of seeing any one factor clearly and distinctively and therefore constitutes as much part of reality as the whole we claim is more fundamental.
A particular view forms when the whole of possibilities is warped by a definite part of it where there is a discrete point between a distinct point and everything generally. And this discrete point is present in every particular point that can be distinct in the general whole of all possible things.
Alph number infinity
From a particular point of view, having an infinity of different variables induces more uncertainty because more unique factors have to be taken into account by a system with limited capacity, but from an absolute universal view having more and more variability of complexity is an antidote to uncertainty because more potential variables are revealed. Let us explore this contrast. the reference frame falls short of disclosing an infinity of potential variabilities because the mere conception of an infinite is a limitation of it to a finitude. How to account for infinity?
You conceive it once at a time, which does not necessarily mean that infinity is cardinal, where you count one and two out of an already established indefinite amount, but that the rate at which infinity is conceived is one at a time, is primarily ordinal. The problem is that the moment infinity is conceived at once, it is contradicted as a one totality, which is only found in the conception of a single particular object. (Point zero energy) the particular object is the order of infinity Into one totality. Empirical science, logic, and all forms of mathematics prove this but showing that in the conception of any particular object, thought, form, is the presupposition of all other.
Empirically this is specifically evident when a single object is said to be formed by an indefinite types of substrucutres. The formula of each thing, is literally everything else, there is nothing specific to the formula of a single thing from a quantum run standpoint. The order by which infinity is conceived is organized into one conception at a time and this means infinity is the indeterminacy of the conception and the world, what is being conceived, is organized in the arrangement where one conception follows another, the content of the order not necessarily matter but the form is always as such where one conception follows right after another. For example, how can a single person go through 6 doors? It is impossible for the single person to go through the 6 doors simultaneously, the only way is to go through each single door one at a time and at some period in time, the person has gone through all 6 doors.
This function where the indeterminacy of the infinite is limited to a finite conception is the only way there can be a sense of outside world, an object for the conception. purposely limited so that its change can account for more variables it was unable to include as part of its conception. In mathematics we may ask; what number is greater the number 1 or 1 billion? The answer really depends on the classification of 1 as ordinal or cardinal. The function of the number 1 to represent a single distinct entity is preluded by the classification of 1 as characterizing 1 infinity because the very first distinct variable that is distinguishable as the initial object for the conception is the infinite.
aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. read aleph-naught or aleph-zero; the German term aleph-null is also sometimes used), the next larger cardinality is aleph-one, then aleph-two etc.
Continuing in this manner, it is possible to define a cardinal number ℵα for every ordinal number α, as described below.
For every 1 countable number is disclosed by 1 aleph ordinal number. So that the number 1000 is disclosed by aleph-one because the aleph-one is infinity of cardinal numbers of which 1000 is only a finite limit abstraction from.
The aleph numbers differ from the infinity (∞) commonly found in algebra and calculus. Alephs measure the sizes of sets; infinity, on the other hand, is commonly defined as an extreme limit of the real number line (applied to a function or sequence that “diverges to infinity” or “increases without bound”), or an extreme point of the extended real number line.
The absolute determination for increasing variabilities provides more certainty as the limit of potential is always approached towards actualization.
1 is capacity for object to conceive itself
(Squared)
Geometry concerns the recreation in the mind of how the object generates as a set of relations in the world.
For example discrete magnitude explains the divisibility of a plain into a number of areas. When a plain is divisible into several areas it does not loose its form but instead is multiplied into a basic unit of measure, I.e, the unit square ascribes the area 1 as the basic unit of measurement. The value of 1 characterizes the form of how the conception remains identical while identifying differently.
The notion for an object to be identifiable as one thing is a result from having implied within it the capacity to be conceived as the unity of parts and not a mere part in a unity. The quality for having a homogenous form is the conception as quality of the fact that distinct things are placed together sharing the same space. Having a value of 1 is the conception disclosing a multiplicity within the same identity. The conception is not merely like vision where a multitude of things are squeezed within the outline of a reference frame, but that the form of each object as it distinguishes itself for perception is the identity of its own self-conception. The magnitude of discrete measure is based on the capacity for a thing to be able to conceive itself. The notion for self-conception does not only mean that the generation of a thing into being is a moment that passes once it has occurred because when a thing makes its way into being it continuous to subsists by taking up a place in relation to the conception of something else, the conception of an object takes up a position in space as well as a place in perception.
The measure of 1 states that no matter of how many variables an object is made up from, the principle of particularity is derived from the capacity for any set of them to be identified as disclosed within single corresponding conception .
Meditate mediate
It is important not to overlook these similarities in language between words as they are not arbitrary. People often over fixate on the specific definition of a words, so they over maintain its peculiar definition by disallowing any speculative connection between words that may portray a similar meaning. It is obvious they “meditate” and “mediate” pertain to completely different things and just to merely conflate them would be a linguistic and logical error. Meditating is an exercise and practice of thinking deeply by focusing the mind for a period of time, it is derived from, meditat- ‘contemplate’, from the verb meditari, meaning ‘measure’; related to “mete” like meter. Meditating is a form of measure by focusing the consciousness on a process.
Mediate describes the activity of mind measures. Mediate or rather ‘to mediate’ means to intervene between two factors and this exhibits the form of connecting link between. Mediate defines the form of a spectrum or a gradient with an element of going back and forth. Interestingly mediate from late Latin mediatus ‘placed in the middle’, as a verb mediare, from Latin medius ‘middle’. In the meaning of mediate, the middle is not a point at the centre of some duration, be it a line or a wavelength, but rather the extent of going back and forth to cover the extent of this motion itself forms the gradient, the ‘middle’ is the duration itself.
Deduction informs subtraction. Number genesis
The act of taking a number from another is an act of inferring particular from general. The act of taking something away presupposes that the source from which it was taken quantifies it.
Inference is conclusion.
The conclusion is the particular and the premise is the general. Mathematical relations follow this fundamental structure of logic. The distinction between the premise and conclusion is not so easy to grasp and is in no way straightforward because what we mean by premise and conclusion relate to the same thing, the argument, and are only different in that they are determinations of their relation. Their difference is not a fixity because they presuppose each other for their own function. In our ordinary thinking we take the determination as coming out, caused, by the object. For example, the conclusion determines the finishing process of the argument. What is overlooked is that the process having as part of its determination the finishing of itself, employs the conclusion as the object of this element. Whereas this may seem as just the mere rearranging of language, it is actually the constructing of thinking, if we assume that words correlate to corresponding ideas.
What is understood as “proofs” employ logic to include the appropriate amount of natural language, which technically admits ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. The philosophy of mathematics is concerned with the role of language as coding the logic implicit in the object. An object in mathematics is anything that is, has been, and could be formally defined, by deductive reasoning and mathematical proofs. The “number” is an object of mathematics because it is used to count, measure, and label, it is the form of these determinations. The “proof” composes the number because it is an inferential argument for proposition. Aristotelian logic identifies a proposition as a sentence which affirms or denies a predicate of a subject with the help of a ‘Copula’. Inferences are steps in reasoning. This does not merely mean moving from premises to conclusions, but that the premise and conclusion explain the nature of determinations that the object defined.
Mathematical proofs are inherently logical arguments.
A proposition may take the form “All men are mortal” or “Socrates is a man.” In the first example the subject is “men”, predicate is “mortal” and copula is “are”. In the second example the subject is “Socrates”, the predicate is “a man” and copula is “is”.
Charles Sanders Peirce divided inference into three kinds: deduction, induction, and abduction. Deduction is inference deriving logical conclusions from premises known or assumed to be true,[1] with the laws of valid inference being studied in logic. Induction is inference from particular premises to a universal conclusion. Abduction is inference to the best explanation.
Axioms may be treated as conditions that must be met before the statement applies. Proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed to be true is known as a conjecture.
A syllogism (syllogismos, “conclusion, inference”) is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two or more propositions that are asserted or assumed to be true.
Ancient Greek philosophers defined a number of syllogisms, correct three part inferences, that can be used as building blocks for more complex reasoning. We begin with a famous example:
1. All humans are mortal.
2. All Greeks are humans.
3. All Greeks are mortal.
The reader can check that the premises and conclusion are true, but logic is concerned with inference: does the truth of the conclusion follow from that of the premises?
The validity of an inference depends on the form of the inference. That is, the word “valid” does not refer to the truth of the (content) of the premises or the conclusion, but rather to the form of the inference. An inference can be valid even if the parts are false, and can be invalid even if some parts are true. But a valid form with true premises will always have a true conclusion.
For example, consider the form of the following symbological track:
1. All meat comes from animals.
2. All beef is meat.
3. Therefore, all beef comes from animals.
If the premises are true, then the conclusion is necessarily true, too.
Now we turn to an invalid form.
1. All A are B.
2. All C are B.
3. Therefore, all C are A.
To show that this form is invalid, we demonstrate how it can lead from true premises to a false conclusion.
1. All apples are fruit. (True)
2. All bananas are fruit. (True)
3. Therefore, all bananas are apples. (False)
A valid argument with a false premise may lead to a false conclusion, (this and the following examples do not follow the Greek syllogism):
1. All tall people are French. (False)
2. John Lennon was tall. (True)
3. Therefore, John Lennon was French. (False)
When a valid argument is used to derive a false conclusion from a false premise, the inference is valid because it follows the form of a correct inference.
A valid argument can also be used to derive a true conclusion from a false premise:
1. All tall people are musicians. (Valid, False)
2. John Lennon was tall. (Valid, True)
3. Therefore, John Lennon was a musician. (Valid, True)
In this case we have one false premise and one true premise where a true conclusion has been inferred.
When we say 1 + 1 = 2, we can say that the conclusion, 2, follows from the premises, 1. The relation 1+1=2 is a valid argument with a necessary conclusion which can in no situation have all true premises and a false conclusion, since there is no situation in which the conclusion is false. We rarely ask how we infer 1 from 1 so as to equal 2 because this is said to be outside the concern of mathematics, since mathematics does not ask why there is a problem, but how the problem is solvable.
The solution of the problem is however only discovered in the very inferential process that gave rise to it in the first place. Any given mathematical object is really the result of the comprehensive logical operations at its roots, and when mathematical objects relate with each other, operating numbers in relation to each other, their logical basis intertwine behind the scenes. The logical concern is whether or not the logical presuppositions of mathematical relations can be brought into consciousness rather than merely remain hidden in the mathematician’s unconscious.
Why can there be only one conclusion in a single argument but many premises? Moreover, where do we infer the premise from if it is prior to the conclusion? These are not only questions of logic but every time a mathematical relation is operated on they unconsciously spring up.
If we say the number 1 is a premise, that means it is a inferential relation. what kind of relation is the number 1? (find here where you derive 1 from 0 and 1 is self-relation)
The conclusion is defined as the finish or the end of process. In an argument the conclusion indicates what the premise is trying to prove and in this way it is the hypothesis of the premise. So how is the end point of an argument also its hypothesis? in other words, if the conclusion does not solidify an answer but only reiterates the question, in what sense is it conclusive? The conclusion is only conclusive in that it involves the sequences that go into the operations of thought.
The conclusion assumes that the beginning is found in the end.
Why is conclusion defined as the finish of the process when it is found in the beginning? This is the case because when the conclusion is stated, there becomes nothing outside of it except the need to justify it, which make up the components that prove it as statement. The very act that the conclusion is proposed presupposes that it is absolute because the act contains the necessary capacities from which it was made, which serve as the basis of the premises. When the conclusion is proposed, it becomes limited to the premises that comprise it. The act is limited to the capacity of its doing (the lifting of my hand is limited to my hand).
The invalid structure is no error but points to the reason why when taken alone the conclusion is universal but in an argument it is particular, and likewise why when taken alone the premise is particular but in an argument it is universal (general). (Above example) there is nothing in the nature of fruit that allows us to infer that bananas are apples. The premise is general shows that in the conclusion there is the capacity for it to contain a false premise. Whereas the conclusion is always true so long as the premises are true, (true premises will always have a true conclusion), that does not exclude the fact that the conclusion may render a false premise therefore a false premise may lead to a false conclusion. This is why a false conclusion is as valid as a true conclusion, not because they are both equally true, but that they counterbalance each other, that one is true because the other is false, that is not the equal to say, the one is false because the other is true, the latter is what constitutes false, whereas the former constitutes true. (but it also doesn’t make sense that just because you are wrong I am right, for you can be wrong while I am also wrong, so what is true must depend on false premise may lead to a false conclusion
the conclusion is general proposition only because it is the hypothesis, in form, but the very statement involves its proofs, these are the premises, which aims to reaffirm the position by ordering the proof in the exact manner it was stated.
the operations of thinking.
because its proposition is particular involves the variables that are its general. Conclusion; Dog is animal, premises dog and animal. The premise is particular on the grounds that it is divided into variables, but the content of the variable, what it is, is a generality.
The conclusion is just the synthetization of the premise. Anything in the conclusion is found in the premise. It’s like saying, 1) cells are living, 2) man is made out of cells, therefore 3) man is living. It would be impossible to conclude that man is not living because he is made out of cells. Since the conclusion is simply the synthesis of the premise.
Now we may ask: is the content dependent on the form or the form on the content. What it means to be “valid” seems to suggests that the form does not depend on the content,
The premise is a statement in an argument that provides support for the conclusion. The Premise in this sense is the previous statement from which another follows. All this means is that the premise indicates the process of the argument, all the possible relations constituting its structure, whereas the conclusion abstracts the totality of these relations into a form. The conclusion is particular because it contains the whole of relations as proposition, whereas the premise is general because it is the proof, the reference which the conclusion refers back to, how the conclusion is inferred.
Having a false conclusion with true premise is impossible because it is like having the parts without the whole, like having all the body parts of a man without having the man that contains all his body parts, or likewise, having the man without the parts that make him up. The conclusion is presupposed by the premise, like the man is presupposed by his parts. The conclusion is really the idea of the argument, the form of it. In other words, the conclusion is the character of the parts, the kind of function they have. A valid argument can also be used to derive a true conclusion from a false premise. We often think that a conclusion is derived after the premise, what we often forget is that the premise in order to be the “previous statement” involves a predicate to that function. In other words, in order for the premise to be the statement providing support, it requires the fact it must prove. The premise itself is inherently a conclusion. The conclusion is not a variable like the premise, but is rather the essence of the variable to be the kind of statement that proclaims to be.
why a whole is a part? A whole is a part because anything outside of what it captures is where it lies as the part.
In the relation 1+1=2, normally, 1 is the premise of 2 and 2 is the conclusion of 1. This structure however is not what we begin with but what we derive at the end of the process.
1 is the conclusion and 2 is the premise, because 2 contains the proof that 1+1, whereas 1 is the proposition that if it is added to itself, it becomes 2. In other words 1+1=2 or 2=1+1, the latter is a “proof” taken as a mathematical fallacy not because it is not true, since it is true that 2 implies the addition of 1 to 1; it is however a fallacy because it is invalid, we add the conclusion prior to the premise, in this way we propose what needs proof as if true prior to demonstrating its truth. This is singular to — Affirming the consequent, sometimes called converse error, fallacy of the converse, or confusion of necessity and sufficiency, is a formal fallacy of taking a true conditional statement and invalidly inferring its converse even though the converse may not be true. If the lamp were broken, then the room would be dark,”) and invalidly inferring its converse (“The room is dark, so the lamp is broken,”) even though the converse may not be true
The fact that 2 is the conclusion for 1 is not entirely correct because in order for 2 to suggest that 1 be added to 1, it must presuppose in the first place that there is a 1 that presupposes itself. 1%1=1, 1×1=1, whereas 2 must presuppose 1 to be the number that makes itself also equal itself. Not only must 1 be equal to 1, but 1 must make every number equal to itself when multiplied or divided.
2%1=2
2×1=2
If we take the conclusion defined as hypothesis of the argument, what it aims to argue or prove, then 1 is the conclusion of 2 because it is what is presupposed in its proof. That 1 ought to have an additive relation with itself. Whereas 2 provides the evidence for 1 because it contains the relation. Therefore, the existence of 2 presupposes the 1 to self-relate to justify itself.
Does this mean we work backwards from 2 to 1 to derive that 1+1=2? Or do we proceed according to our common sense from 1 to 2?
When we count we begin with 0,1,2,3,4… This is actually an informal way contrived into a formal axiom concerning the supposition of how numbers naturally proceed. The reasoning behind this is as follows; when we presuppose 0 as the initial number of the series, we technically presuppose infinity, which is said to be quantitatively the largest possible number. The aleph numbers, for example, aleph-zero are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. The aleph number is made to represent infinity into a finite variable while retaining the function of infinity. This allows, for efficient technical reasons, to derive from infinity every possible relation. The most basic set of arithmetic begins with 1 after 0 because that is actually the abstracted variable relation closest to infinity.
Qualitatively 1 is the closets to infinity. 1 is the contradiction.
One, sometimes referred to as unity,[2] is the first non-zero natural number. It is thus the integer before two and after zero, and the first positive odd number.
Any number multiplied by one remains that number, as one is the identity for multiplication. As a result, 1 is its own factorial, its own square, its own cube, and so on. One is also the result of the empty product, as any number multiplied by one is itself. It is also the only natural number that is neither composite nor prime with respect to division, but instead considered a unit.
1 is the quality of form and not merely a quantity
This means that 1 involves more relations. It is found in every number.
The farthest off we go from 1 the more finite and definite the relations become. Really 1 is the end of the series of finite variables and not merely the beginning. This means that if we count every number, the end result is 1.
1 is premise of 2 because it is a necessary component of it. What then is the difference between premise and conclusion?
A premise is a proposition from which another is inferred and follows as a conclusion. 1 is inferred from itself.
This means that the very presupposition of 1 is its inference. The presupposition is not arbitrary but must be derived, where it is derived is what presupposes it. And therefore the inference is based on the notion that there is reason to presuppose, why else would you presuppose anything?
This means we presuppose 1 from itself when itself is what is being presupposed. This is how we infer the conclusion. The mathematical function of the inference is literally informed by the infinite regress: defining something in terms of itself. The premise is general in this very sense, that it infinitely infers itself. For example, this is how we can have 1000 from 1. The difficulty arises in the fact that the conclusion is particular. When we say that 1-1=0 or 1+1=2, the conclusion is a relation. How can something derived from itself result in a conclusion that contradicts itself? In other words, if 1 is derived from 1, its relation, 2 is something other than itself.
(explain how 2 is particular because it merges by sublation the inference of the premise into a specific manner. That if the premise is general in the sense that it is infinite, the conclusion derives from that infinity a particular relation, but does so logically in the sense that, chronologically in the sense of series, then randomly.
(Add here that infinite regress is feature not contradiction)
Mathematical logic
Mathematical logic appropriates the dialectic in the most abstract sense. Hegel’s dialectic exists in first order mathematical logic in the sense of a “binary function”. In every first order mathematical language you need a binary function. A binary function is defined in this sense: derive a unity from a duality, derive 1 from 2. it takes in two elements and produces one.
1 is greater than 0 is true under the greater than function: the formula mathematically is:
1>0 —-> truth. 0 is greater than 1 produces false: 0>1—–> false. This relates to the dialectic in the sense that two binaries produces unity. In every mathematical language you need two aspects under a binary function to produce unity. This is the equation explaining this:
F:G*G——>H
F: is a rule that assigns to every element of the domain G*G a unique element of the co-domain H
G: is a set, and a set is a collection of objects, like 12345 etc.
H: is any set, it could be G or not G.
This formula represents the dialectic, in this way the dialectic manifest it-self into first order mathematical language.
The dialectic in the form of binary function does not assum that every truth it self has it’s own independent being, but it is related to the same being, that is, what it means to be independent being is a relation of a thing with itself, the unity is a gradient.
11#- Of the logical relations of math
Mathematics is the science of quantity and logic is the science of quality. For Peirce as well as for Hegel and Aristotle, logic is more fundamental than mathematics. (Quality is more fundmental than quantity because quantity is itself a quality, that it is a quantity. A quality is however a quantity because that is its measure) Peirce says that logic shows all the possibilities of a calculus before the fastest route can be taken. For Aristotle, logic belongs to the system of metaphysics, the first philosophy, which is more fundamental then math, or that, mathematics is a particular branch of logic dealing with the external relations of principles. Hegel elaborating Aristotle sought that logic is the organic way matter takes on form and moves.
Each number is an abstraction of a relation that predicates it:
1 is the loneliest number because besides it is nothing else but 0. The 0 is the other of 1, that which is less than 1. The number 1 therefore takes this negation, the 0, as the decimal place whereby to identify itself. The number 0 is therefore the place where 1 perceive itself and reaffirms itself as the number 1. The number 0 is the principles of identity in relation to the 1. This is proved by the elementary operator of the negative (1-0=1). The reason why 1 negative of 0 is equal to 1 is because of the more fundamental principle of identity, in that 1 in negation to nothing, or lack-of-1, is just the reaffirmation of itself as the number 1.
The 1 therefore identifies itself in that lack of itself, and it does this in order to confirm its existence, otherwise without this reaffirmation, the 1 would be a 0, which cannot be true, because then the number 0 is an entity denoting a quantity of 1. The number 0 in its own identity requires it to be a quantity of 1.
would be now a 1 and a 0, bringing us back to the initial place. This is why the following relation is thusly: 1+0=1, because 1 in addition to nothing is just the reaffirmation of itself as 1. But in this affirmation we have the number 2. The number 2 is 1 in recognition of itself as 1, that is 1 in addition to 1 (1+1=2). This recognition is an addition because it is a knowledge of itself as opposed to 0. Now if we multiply 1 with 0 we have a reaffirmation of 0 (1×0= 0). The reason is that when 1 multiplies 0 it is just reaffirming the number 0 and not itself. But if the number 1 multiplies itself, the result is another number 1 (1×1=1).
Logic and math- after and before
This far we have been stating that logic, according to Hegel, is concrete, but what does this exactly mean? It means each logical component derived from the object that makes it tangible has an existence in sequence to the proceeding logical propositions. In mathematics, the Aristotle principle of a “continuity” is that each continuum has its own limit. The limit in each series has a qualitative measure in that it is greater than all previous numbers and less than the numbers that follow. In logic, each limit itself bears a particular kind of qualitative measure in relation to other limits. When the numbers after are greater than the before, the numbers before are greater than the numbers before them, this is an example of logical continuity in mathematics, because the sequences is connected by the common quality of the greater denominator.
Each logical proposition is a specific kind of component integral to the whole. But the whole is but just the next logical proposition following the previous, since it cannot be both the later and the before as they both contradict each other is what makes them into parts, the later is in part greater, while the former is in part lesser. Each logical proposition is the whole for the next part. Peirce indicates the axiom that: the whole is greater than the parts: is a theorem for finite numbers not infinite. In physics, nothing is created or destroyed- and this principle of logic translates into the essential principle of matter- matter cannot be destroyed or created but altered. In logic every proposition is a presupposition that is to be proved, that is to say, the content of logic has nothing to waste away, because it has no specific content so that anything logically conceivable is truly the content for consideration, or in other words, the content of logic, is always present for logic, it does not waste away.
This is one similarity that logic shares with mathematics except in math, the proof is derived by way of the fastest steps or shortest rout whereas in logic because each possibility is considered in the proof, the more strenuous and prolonged the calculus the more logical, the more possibilities considered. This kind of analysis is for example seen on the works of Aristotle where all of his books have the common theme of laying out all possibilities that go into a phenomena or a thought process, some of which appear contradictory, but it is these contradictions that fill out the comprehensive analysis that go into any particular fact or idea. But mathematics itself presupposes logic, math is in fact a possibility of logic itself, that the fastest way to prove any proposition is itself a logical principle.
When we are talking about the nature of logic as concrete, what we are essentially saying is that each logical proposition is an actual object which takes on a tangible reality. For example, Hegel states that the Notion is the property of logic that makes it concrete. The Notion for example is the property for what Peirce identifies as Deduction, Induction, and hypothesis (explain). The notion works by the ‘negation of negation’ (look above).
An example of logical proposition that takes on a concrete reality in any tangible object is this:
Hegel works backwards from the object to derive its logical propositions. The logical deconstruction of the object is a very tricky task. Hegel proceeds inductively from any tangible object and derives general rules of logical from that object that is relevant to all objects. But be careful not to confuse the matter in that the logical rules are not the parts making up the object it is rather that the object is the part from the logical. The object is the part and the logic is the whole. Remember that each limit is the whole for the next.
To say that each logical proposition is a part making up the whole is applying the the axiom “that the whole is greater than the parts” which in this case, the whole is confused with the parts. The whole is not the object, but rather its logical propositions. Simple immediate being is the whole for any following logical proposition because it’s actual existence exists as such- immediate being is just simply is without being anything specific. That of course has a negation relationship with universal being- which is everything that can be- which in relation they produce that which is. The principle of identity is immediate being.
“Being is simple as immediate being; for that reason it is only something meant or intended and we cannot say of it what it is; therefore, it is one with its other, with non−being. Its Notion is just this, to be a simplicity that immediately vanishes in its opposite; it is becoming. The universal, on the contrary, is that simplicity which, because it is the Notion, no less possesses within itself the richest content.” (Logic 1327)
The universal is therefore free power; it is itself and takes its other within its embrace, but without doing violence to it; on the contrary, the universal is, in its other, in peaceful communion with itself. We have called it free power, but it could also be called free love and boundless blessedness, for it bears itself towards its other as towards its own self; in it, it has returned to itself. (Logic 1331)
The relation of negation of negation takes on a positive form. The positive logically speaking is not a variable separate from the negative, it is rather the relation of the negative with itself that assumes a positive nature. This is a deductive conclusion and so the positive takes on an entirely new nature on its own different then the negative relations that gave its existence. What this means is that the inherent nature of the negative proposition in logic always bears a positive existence, and so the negative proposition is not itself the lack of existence, it is instead the negation of any existence, including the non existence, the negation of the non existence is existence. It is even the negation of itself when it does not exist. It is the negation for its non-existence.
In nature the negative logical principle takes on the concrete form of time. Time is the example of the activity in its negative nature. It is devoid of any content but yet active and intentful. Towards what time is active is the responsibility of creativity. creativity is the opposite and positive principle of time. According to whitehead creativity is operative in the same as time but it is the content of the activity. Creativity does not mean spontaneity as we understand it in the arts. It is rather the ability of synthesizing its logic into a unique proposition. Creativity manifests as life in nature- producing an infinite diversity of itself. Evolution is its process. Whereas time manifest as death. Death is the sublation of time and life the sublation of creativity. Creativity exists in time as time exists in creativity. Likewise death exists in life as life exists in death. They exists in each other inversely, meaning that life is the positive logical principle whereas death is the negative. The nature of life is positive means that it exists in every kind of possible form- it takes on an infinite set of unique forms.
The nature of death is negative and like time, it is induration in activity without content, it is the activity for the sake of the activity. So death in relation to life itself becomes an organism. The relationship between life and death is an organism implicit inside an organism. Each living unique form possess with it the same organism. Death is the one organism implicit in all organism. Death is the one in the many. Death understood only as corruption does not fully indicate the purpose it serves because death is the very limit of creativity to bring about the infinite forms of life. Otherwise life would be reducible to time without content, which on its own would just be the duration of nothing, which as we shown above, is impossible because non-existence bears a negation of itself therefore producing existence.
(Michaela Reyes )
The application of logic to math is one often ignored because logic and math presuppose opposite yet related tasks: logic is meant to derive every possible outcome of a proposition whereas math aims to solve the the proposition as efficiently as possible by taking the shortest amount time with the least amount of steps. And so in some sense math itself presupposes logic as its principle- in that the shortest route to solving a proposition is one that is logical. Yet logic at the same time must apply math so as to explain every possibility associated with a logical proposition- among the primary possibilities is the quantitive measure that math has as its content. The two are as necessary for each other as much as they are different from one another.
And although Perice emphasizes their difference he nevertheless applies them together in their own respective right- that is why to a degree he is such a successful mathematician as he saw the right application of math is a logical application, while most mathematicians where not very logical. The greatest challenge that mathematicians face is dealing with the concept of infinity. This concept can only be made sense of by including logic in the application of math.