Foundations of maths

[DevilSolution]

I created a similar topic regarding math and logic but my initial OP was totally incorrect and secondly this delves a little deeper into other factors of math.

 

The simplest way to express maths is to postulate the existence of a single value.

Once we have a value we then postulate 2 basic operations that this value can use, addition and subtraction (multiplication being a function of addition and division being the abstract inverse of multiplication, powers being extension of multiplication and root being extension of division (specific forms of that operator)).

 

From here we have an infinite set of natural numbers using just addition and once the other operators are used we have an infinite set of real numbers.

 

Having summed maths in this form it may be best to explain axioms(as we have used atleast 3) and proofs.

 

 

Okay so firsly we have axioms / postulates which are simply propositions of truth.

We then try to apply some form of proof onto the axiom which gives it its logical conclusion (true or false). Now when trying give a proof we use a basic set of principles with which it must successfully navigate to be determained as true. The primary forms of proof to my knowledge are:

 

The counter example (finding an error in the axiom)

 

The direct method (minimize the proof to some expression thats always true)

 

The indirect method (reduction to the absurd, inverse the postulate)

 

The method of mathematical induction (the domino effect, if the first expression of a sequence is true, then it follows the rest are)

 

 

At this point its important to declare the use of logic. The only real value an axiom holds is its logical outcome from the proof. Not only that, pure logic is used as the method of proof. (Which is my argument for math being a product of logic).

 

 

Now i would like to analyse a few aspects of the process weve used so far then offer some questions regarding the fundamental nature of maths.

first of all in my example i postulated that the value 1 exist. The problem with the proof is that logically we must accept its existence. Secondly i would like to know whether its possible to postulate operators in the manner i did? Axiomatically creating the operators using logic is possible as is proven with the constructs of "and", "or" and "not" which are equivalent to all the operators when combined in a particular form (again further adding to the concept that all math is formed from logic).

 

So my initial question is: How are numbers and operators related? If the operators already exist within the realms of logic, does the addition of the axiom 1 therefor create mathematics?

 

 

I would also like to know about the logic we personally apply to mathematics, such as BODMAS.By default there is no logical reason to apply precedence to operators other than for our own abstract use of it.

What i mean is are we defining the laws of maths ourselves? Hence making our form of maths only a branch of logic.

 

Also im interested in understanding the abstract nature of maths vs physics. For example in the mathematical realm we use the abstact concept of a circle, its used as a fundamental aspect of maths, not only within the area of geometry but it also extends into other branches of maths such as statistics and probability. However in the physical world a circle doesnt exist yet we apply the various uses of pi to the physical world, why does a purely abstract mathematic concept have such importance in the real world? and how does one cross the barrier of abstract maths into the physical world?

 

One final question, in regards to the "proofs" we use, is it possible that theres errors in these processes? Could there be an ultimate logical expression that alone stands as proof. In other words instead of having multiple methods as proof could there be a single expression that proves the axiom?

 

One final note, it seems that maths is directly related to the physical world, we purposely use it for its functionality within the physical world. However it seems we have various aspects of the mathematic realm which has no bearing on physical reality (for example the use of dimensions in maths, which geometrically pile up, in comparison to our 3 dimensional world of x, y and z. Here the maths doesnt relate to reality). So is maths an extension of physics? Or is it the physical world that dictates the use or usefulness of math?

 

Regards and sorry for such the long post.

Edited September 8, 2015 by DevilSolution

[studiot]

 

The simplest way to express maths is to postulate the existence of a single value.

Once we have a value we then postulate 2 basic operations that this value can use, addition and subtraction (multiplication being a function of addition and division being the abstract inverse of multiplication, powers being extension of multiplication and root being extension of division (specific forms of that operator)).

 

 

 

Why start with a 'value' and why does existence need to come into it?

 

We define operations, not postulate them.

 

Definitions come first then postulates come after.

 

Postulates refer to something more fundamental than value.

 

Postulates refer to the relationships between defined objects. Preferably one relationship per postulate.

 

A good example of a relationship is connectedness.

 

For instance in a triangle vertex A is directly connected to both other vertices B and C, but in a quadrilateral it is only directly connected to B and D but not directly to C.

In fact the triangle is the only polygon for which every vertex is directly connected to every other.

 

You can build up a whole branch of mathematics based on this.

[DevilSolution]

 

Why start with a 'value' and why does existence need to come into it?

 

We define operations, not postulate them.

 

Definitions come first then postulates come after.

 

Postulates refer to something more fundamental than value.

 

Postulates refer to the relationships between defined objects. Preferably one relationship per postulate.

 

A good example of a relationship is connectedness.

 

For instance in a triangle vertex A is directly connected to both other vertices B and C, but in a quadrilateral it is only directly connected to B and D but not directly to C.

In fact the triangle is the only polygon for which every vertex is directly connected to every other.

 

You can build up a whole branch of mathematics based on this.

Im aware of how postulates work, in reference to defining operators instead of postulating them, do they all come from logic as i proposed? such that addition is simply a&&b etc

 

Also if we dont somehow postulate that a value exists, where else would we get values from? Its a fundamental axiom that a number must first exist.

Edited September 8, 2015 by DevilSolution

[studiot]

Depends what you mean by logic.

 

You have to start with definitions.

[DevilSolution]

Any form of propositional, sequential or combinitorial logic thats defined by the operators "and", "or" and "not". These allow for things like addition and subtraction and as most operators are derived from these any operator could be defined in logical terms.

[studiot]

Any form of propositional, sequential or combinitorial logic thats defined by the operators "and", "or" and "not". These allow for things like addition and subtraction and as most operators are derived from these any operator could be defined in logical terms.

 

Don't you think there is anything else to mathematics then?

[ydoaPs]

Why start with a 'value' and why does existence need to come into it?

 

We define operations, not postulate them.

 

Definitions come first then postulates come after.

 

Postulates refer to something more fundamental than value.

 

Postulates refer to the relationships between defined objects. Preferably one relationship per postulate.

 

A good example of a relationship is connectedness.

 

For instance in a triangle vertex A is directly connected to both other vertices B and C, but in a quadrilateral it is only directly connected to B and D but not directly to C.

In fact the triangle is the only polygon for which every vertex is directly connected to every other.

 

You can build up a whole branch of mathematics based on this.

I agree with this. Why should we start with values rather than something more general like sets (or even more general than that, and use categories)?

 

You can start with objects A, B, and C, as well as processes x:A->B and y:A->C

 

[A]->

|

\/

[C]

 

And then you can define general division in terms of sections and retractions. If you're wanting to find foundations (which, as a philosopher in an HPS program, I think are unlikely to be found), then shouldn't you go as generally as possible?

Im aware of how postulates work, in reference to defining operators instead of postulating them, do they all come from logic as i proposed? such that addition is simply a&&b etc

 

Also if we dont somehow postulate that a value exists, where else would we get values from? Its a fundamental axiom that a number must first exist.

If you want to use logic, addition is actually defined as the disjunction of sets, not their conjunction.

 

Zero is the null set. One is the set containing only the null set. Two is the null set disjoined the set containing only the empty set (or the set whose only two members are the empty set as well as the set containing nothing but the empty set). Three is the set whose three members are the null set, the set containing only the null set, and the set containting only the null set and the set containing only the null set. And so forth.

 

0={}, 1={{}}, 2={{}, {{}}}, 3={{}, {{}}, {{}, {{}}}}, etc.

first of all in my example i postulated that the value 1 exist. The problem with the proof is that logically we must accept its existence. Secondly i would like to know whether its possible to postulate operators in the manner i did? Axiomatically creating the operators using logic is possible as is proven with the constructs of "and", "or" and "not" which are equivalent to all the operators when combined in a particular form (again further adding to the concept that all math is formed from logic).

 

So my initial question is: How are numbers and operators related? If the operators already exist within the realms of logic, does the addition of the axiom 1 therefor create mathematics?

It sounds like what you want here is the Peano axiomatization.

[Scott Mayers]

For clarity, I'm numbering your questions, DevilSolution.

 

(1) So my initial question is: How are numbers and operators related? If the operators already exist within the realms of logic, does the addition of the axiom 1 therefor create mathematics?

 

 

A number often act as 'premises' to the system of logic defined by its operators. However, numbers and operators can both be interpreted as the distinction between 'states' and 'actions' akin to our use in language of noun-like entities and verb-like ones. Both can be interpreted in terms of the other just as grammar can find any form of a noun-entity as a verb and visa versa. For instance, the verb, to 'walk', has a noun-version of it, "walking" (we call a gerund). Often some logics to prove math treat the initial premise of a number based on a unit and a zero state only. Then all other numbers are derived from these. As such Zero and the Unit are somewhat treated as logical entities that are the only postulated ideas and try to avoid being thought of as 'numbers' per se.

 

I think these two elements are also something that can be 'empirically' observed in nature by each individual and why these are often easiest to presume. Then the logical elements of 'operations' are also on par with numbers in that they are simply assumed as relations on one or zero of between any of the two of them together as a relation. The 'complement' is first defined based on the domain of each element. For only two elements (Unit and Zero), the complement of one is simply the other. [note that I read later by another poster who assumed that addition was identical to logical 'or'. This is an error of interpretation based on the fact that many of the same symbols get used in both.

 

Comparing any two elements defines the possible distinct relations they have.

 

0 relates to 0 in that they are the same and also lack content [Property of Zero as constant]

1 relates to 1 in that they are the same but have content [Property of One as constant]

 

Both of these relate to each other in that they are "the same" yet differ in their semantic assignments," one having content and the other lacking it". So the relation of both the above is sometimes called, "coincidence" [Property of Coincidence (Same value to same value)]

 

0 relates to 1 in that beginning in Zero complements 'ending' in a Unit [Property to Change from Zero]

1 relates to 0 in that 'beginning' in a Unit 'ends' in a Unit [Property to Change from a Unit]

 

Collectively, these two relations define the complement here (Negation for binary.) [Property of the Complement]

 

Notice how the first two are initially 'static' while the latter two are 'dynamic' (vary from whichever you begin on). As such, the first two get treated as the idea of 'number' where the other two are treated as the idea of an 'operation'. But we also show they only "relate" by considering all optional ways to compare given a 0 and a 1. The first two can also be described as the 'dynamic' operation of 'staying the same'. Yet we often do not choose to think of it this way arbitrarily.

 

These can be variably interpreted. The first two can be interpreted as the 'dynamic' relation of "staying the same" and the second two as "to change". Since we default to the idea of "consistency" as humans, we often ignore the first two as 'dynamic' though we CAN. For instance, if we define the "changes" in the second two first as "negation" and use the sign, '-' which reminds us of a sideways "one", then we might define the "non-changes" of the first two as "positions" and use the sign '+'. This is like crossing out the sideways "one" to show opposition. Some may even interpret this a "zero-like" concept in that it is an action that requires no change. Thus one might think of these 'operations' as numbers too.

 

I can go on in quite depths here. But this is enough to get your mind going on how logic originates and eventually lead to all logic and math. However, this should answer your first question. The 'number' and 'operation' concepts are actually equally interpreted as derived from each other depending on your personal choice to begin with. An operation can be interpreted as a 'dynamic number' or a number in a different "direction", "dimension", or "perspective". OR you can interpret a number in terms of the operations as I showed above. The act of "staying the same" can be what '0' means and the act of "changing" as '1'. Note how the very symbols we use hint at how the ancients likely used the zero as a closed loop and the symbol for one as an open (non-closed or incomplete) idea, like a fence that cannot close in your yard allowing your cattle to escape. These are a dynamic interpretation and also ironically opposes the idea that we find comfort in thinking of "closed lines" as finite and more real as states and "non-closed lines" infinite.

 

I hope this answers your question.

 

 

(2) I would also like to know about the logic we personally apply to mathematics, such as BODMAS.By default there is no logical reason to apply precedence to operators other than for our own abstract use of it.

What i mean is are we defining the laws of maths ourselves? Hence making our form of maths only a branch of logic.

 

 

I think people differ on answering this dependent upon their experience. But I believe the laws of logic/math are real and all that is actually real but is hard for many to accept. This is the biggest conflict I have with our present paradigm with regards to "science". "Science" itself is a term derived from meaning what is based upon one's senses. However, from the previous question, you should be able to see that the ideas themselves are "observed" as you think of logic itself. The laws of logic can be inferred by experience as absolute generalization relating necessarily to ALL possible "observations" and why I interpret logic and math as real rather that simply as 'tools' without attending to whether they are real or not in science.

 

But both interpretations ARE what exists among people of most times trying to make sense of our reality. And both CAN be valid approaches to making sense of reality. I take this as a third interpretation. Thus we need to accept both logic/math as real as well as the contingent experiences using our senses to induce a general set of 'laws'. Those precedence laws are not actually 'arbitrary'. The particular approach as humans to communicate logic can define logics that appear distinctly different but they all actually connectively relate in that one system of logic can define another separate system. For instance, the logic using operators of (position), "negation", "and", or "or" have to create or derive math as a secondary or later logic. This was my point I mentioned about a poster later thinking that addition is the same as "or". In computer logic, "addition" is defined using an "exclusive or" and the "and" operator. It gets even more complex in that it has to define things like memory spaces (or registers) and control behaviors like moving data (like "store" or "load").

 

 

(3)Also im interested in understanding the abstract nature of maths vs physics. For example in the mathematical realm we use the abstact concept of a circle, its used as a fundamental aspect of maths, not only within the area of geometry but it also extends into other branches of maths such as statistics and probability. However in the physical world a circle doesnt exist yet we apply the various uses of pi to the physical world, why does a purely abstract mathematic concept have such importance in the real world? and how does one cross the barrier of abstract maths into the physical world?

 

 

I disagree with the concerns of many to dismiss abstractions or "forms" of logic, math, or geometry as being unreal as I mentioned before. I further believe that ALL physics are dependent upon logic and the physics are just our contingent experience of the universes (or totality's) by perspective only.

 

 

(4) One final question, in regards to the "proofs" we use, is it possible that theres errors in these processes? Could there be an ultimate logical expression that alone stands as proof. In other words instead of having multiple methods as proof could there be a single expression that proves the axiom?

 

I believe it is about 'errors'. As entities in reality, it is hard to find a universally agreed upon method other than how we do so through political acts between people communicating these ideas. I believe there definitely IS an ultimate logic that encapsulates all other rationalizations in reality with respect to nature or totality itself. It is just difficult to explain. But I believe this CAN be done.

[Acme]

...

I believe it is about 'errors'. As entities in reality, it is hard to find a universally agreed upon method other than how we do so through political acts between people communicating these ideas. I believe there definitely IS an ultimate logic that encapsulates all other rationalizations in reality with respect to nature or totality itself. It is just difficult to explain. But I believe this CAN be done.

Axioms by definition are not subject to proof and Gödel's incompleteness theorems prove there is no universal/ultimate logic, i.e. axiomatic system.

Gödel's incompleteness theorems @ Wiki

Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible, giving a negative answer to Hilbert's second problem. ...

While you may argue that my choosing Wiki is a political act, one can find any number of other sources for this information. Incompleteness is as incompleteness does.

[Scott Mayers]

Axioms by definition are not subject to proof and Gödel's incompleteness theorems prove there is no universal/ultimate logic, i.e. axiomatic system.

Gödel's incompleteness theorems @ Wiki

 

While you may argue that my choosing Wiki is a political act, one can find any number of other sources for this information. Incompleteness is as incompleteness does.

I've discussed this interpretation before elsewhere. Words like assumption, premise, axiom, postulate, among others all depend on the particular use of the author using them and normally mean the same for simple uses. But if you want to be more descriptive, especially where you want to distinguish between initial inputs to an argument or form to contrast specific kinds of inputs based on assumptions within a system of arguments, then it is useful to assign more distinct meanings. I like using "postulate" to those assumptions we input at the beginning of a system of arguments and axioms as input assumptions based on prior theorems or its own postulates or definitions of that system.

 

I'm very familiar with Godel's incompleteness theorem and have contention with it as it allows a set to be defined such that it's universal description can be included within the set. [A Set of all Sets, for instance which requires it to be included in itself] Implicitly Godel theorem is what I think tipped a change in science but it is not valid to your interpretation although most assumed it as such and digresses to more specifics worthy of discussing separately from this thread.

 

I'm not sure why you'd think that mentioning this or Wikipedia was what I'm thinking of by 'political'. As to Godel's theorem and the surrounding philosophy and science of the day, this involved a lot of politics with regards to the world wars, the standards of encouraging students into science, the institutions' pressures to appeal to threats by and for religion as opposed to atheism, etc, etc.

 

But my response was to the OP and he/she sounds too new to this to get to such depths at this point.

[Acme]

I've discussed this interpretation before elsewhere. Words like assumption, premise, axiom, postulate, among others all depend on the particular use of the author using them and normally mean the same for simple uses. But if you want to be more descriptive, especially where you want to distinguish between initial inputs to an argument or form to contrast specific kinds of inputs based on assumptions within a system of arguments, then it is useful to assign more distinct meanings. I like using "postulate" to those assumptions we input at the beginning of a system of arguments and axioms as input assumptions based on prior theorems or its own postulates or definitions of that system.

As others pointed out to you elsewhere, it's not up to you to redefine terms for everyone else.

 

I'm very familiar with Godel's incompleteness theorem and have contention with it as it allows a set to be defined such that it's universal description can be included within the set. [A Set of all Sets, for instance which requires it to be included in itself] Implicitly Godel theorem is what I think tipped a change in science but it is not valid to your interpretation although most assumed it as such and digresses to more specifics worthy of discussing separately from this thread.

Well, contend all you want. If you can disprove Gödel then you will be famous.

 

I'm not sure why you'd think that mentioning this or Wikipedia was what I'm thinking of by 'political'. As to Godel's theorem and the surrounding philosophy and science of the day, this involved a lot of politics with regards to the world wars, the standards of encouraging students into science, the institutions' pressures to appeal to threats by and for religion as opposed to atheism, etc, etc.

 

But my response was to the OP and he/she sounds too new to this to get to such depths at this point.

Mathematical proofs aren't political. It doesn't matter who or what your response was to; what matters is that it was in error.

[Scott Mayers]

As others pointed out to you elsewhere, it's not up to you to redefine terms for everyone else.

 

Well, contend all you want. If you can disprove Gödel then you will be famous.

 

Mathematical proofs aren't political. It doesn't matter who or what your response was to; what matters is that it was in error.

Many of you need to go back to school to take logic proper. I also recommend a need for more philosophy requirements to graduate for even a Bachelor's degree. On definitions, my point, as it relates to philosophy, is that you have to attend to the particular people you discuss things with to find common agreed to definitions. You dis'ed me for your fixed interpretation of the word "axiom" in your own head. To anyone versed in dialectic, you have to begin with definitions up front for the present argument. It is impossible for any one person to be sufficiently flexible to speak in everyone's different jargon they prefer to use in different fields. A "postulate" is an old term most often used in the Greek geometry and focused on procedural drawings (like using a compass and straight edge) to demonstrate the math without using numbers because they lacked our present zero in their language, whereas in later evolved maths using the term, "axiom" for assumptions, this DID use numbers. But if you've read across different literature, you'd see that most use a variety of different terms often to specifically reference their intentional meaning.

 

As to Godel (excuse that I'm not bothering to use the character map for the 'o'), he even had doubts of it later as new set theory ideas came about that limited the dilemmas that lead to Godel's Incompleteness Theorem. He also refused to allow for a meaning to infinite actions to be de-fined (of finite meaning or closure). This defaults his theorem by definition begging. But the political use of his theorem was intended to aid in demarcating science from other socially accepted views from those who either practiced fraudulent science or the religious fervor of the West who used logic in apparently arbitrary ways to appear to 'prove anything'. This was more about expedience, not sincerity to truth. Also, the Big Bang theory was desired politically over the Steady State model in order to appeal broadly to all people if only to keep the average yokel from politically succeeding to attract students to science.

 

The math itself isn't political, it is the politics that favored those who would present theories that don't insult religion, economic industries, governments or the authorities of distinguished universities who desire controls. The theorem is political in that it is one of the major theories that actually turned the tide to foster a different educational approach. Fears of Communism with their 'atheistic commie' thoughts were in the minds of most back then. And the World Wars that were based on the very logical analysis that also fostered the growth of science was suddenly recognized to be a threat if not contained in a way by the politics. Science first and foremost to politics is the factor that gave them the weapons of mass destruction. These things were sufficient motivators to initiate a clear motive to contain science and distinctly separate it from the foundational reflective training except for the privileged ones who could earn their trust.

 

As such, they succeeded as even here and in the split thread can be witnessed an odd 'faith-like' blind belief in a specific method without sincere self-referential experience in scientific philosophy. Nor am I being absolute here. But it is those like you how yell the loudest and feign that scientists are all on the same side. I'm glad to see SillyBilly here on this site and will be joining him in his efforts as I get where he's coming from.

 

Philosophy 101 lessons. From Socrates on: Define the terms and question everything! This doesn't mean define a distinct set of terms in stone either. It means that in discussing anything within any group of people, you negotiate the definitions even if they are to be used only specifically for the argument at hand. The question everything with openness and sincerity without insulting others (Charity to grant others by default as being sensible at all times). Rhetoric (like the informal logic dealing with fallacies in conversational language), a subdivision of logic, is more lacking in pure scientists than most other areas. I discover more students who learn of this now from the light discourse on this in those like Richard Dawkins or a multitude of others who appear to have only recently popularized this. It's about a means to converse in a socially clear way or with precision as good lawyers or politicians are more versed in.

 

So yeah, it is about me (and you) to redefine terms where necessary to communicate better to meet in the middle.

[Acme]

Foundations of Mathematics > Axioms > @ Wolfram Mathworld

An axiom is a proposition regarded as self-evidently true without proof. The word "axiom" is a slightly archaic synonym for postulate. Compare conjecture or hypothesis, both of which connote apparently true but not self-evident statements.

[Scott Mayers]

Foundations of Mathematics > Axioms > @ Wolfram Mathworld

I don't get your contention with me on this???

 

This only reminds me of an elitist antique book seller I knew years ago who was so absurdly pedantic as to the specific pronunciation of "Principles of Mathematics" (Russel and Whitehead's three-volume tomb) when I inquired about whether he had it. I pronounce it as "prinsiples" just as we use it colloquially in my world and he insisted adamantly that I pronounce it "prinkiple" as if it had necessary significance. This is the same with your insistence on particular use and is why and where the image of many scientists get appropriately labeled as elitist (or snobbish). Who cares which words we use as long as when or where we use them we can relate. Are you not flexible enough to be able to interpret the sign "+" for meaning "inclusive OR" in logic because you only think of it as meaning "addition"?

 

And no, I didn't buy the book. It was like $15, 000 as some collectible original. I opted to read the ones from my library.

Edited October 8, 2015 by Scott Mayers

[studiot]

 

Many of you need to go back to school to take logic proper. I also recommend a need for more philosophy requirements to graduate for even a Bachelor's degree. On definitions, my point, as it relates to philosophy, is that you have to attend to the particular people you discuss things with to find common agreed to definitions. You dis'ed me for your fixed interpretation of the word "axiom" in your own head. To anyone versed in dialectic, you have to begin with definitions up front for the present argument. It is impossible for any one person to be sufficiently flexible to speak in everyone's different jargon they prefer to use in different fields. A "postulate" is an old term most often used in the Greek geometry and focused on procedural drawings (like using a compass and straight edge) to demonstrate the math without using numbers because they lacked our present zero in their language, whereas in later evolved maths using the term, "axiom" for assumptions, this DID use numbers. But if you've read across different literature, you'd see that most use a variety of different terms often to specifically reference their intentional meaning.

 

 

With the greatest respect, perhaps you need to consider taking your own advice.

 

The (ancient) Greeks did indeed start with definitions.

But they then introduced a set of statements they called postulates and what we now call axioms.

They also introduced a second set of statements they called common notions and we also call axioms today.

 

All three of these types are statements made without proof.

The Greeks distinguised postulates as those special to the discipline under consideration and common notions as those with wider more general application beyond.

 

We call both of these axioms because we start with the most general and say that the common notions are induced from the more general.

 

What you have described above as postulates they called propositions.

Propositions are statements deduced from the original statements by some system of proof.

Today we call these theorems and lemmas.

Edited October 8, 2015 by studiot

[Acme]

 

Foundations of Mathematics > Axioms > @ Wolfram Mathworld

 

An axiom is a proposition regarded as self-evidently true without proof. The word "axiom" is a slightly archaic synonym for postulate. Compare conjecture or hypothesis, both of which connote apparently true but not self-evident statements.

 

I don't get your contention with me on this???

 

You said we needed a definition and I gave it.

 

I've discussed this interpretation before elsewhere. Words like assumption, premise, axiom, postulate, among others all depend on the particular use of the author using them and normally mean the same for simple uses. But if you want to be more descriptive, especially where you want to distinguish between initial inputs to an argument or form to contrast specific kinds of inputs based on assumptions within a system of arguments, then it is useful to assign more distinct meanings. I like using "postulate" to those assumptions we input at the beginning of a system of arguments and axioms as input assumptions based on prior theorems or its own postulates or definitions of that system. ...

Many of you need to go back to school to take logic proper. I also recommend a need for more philosophy requirements to graduate for even a Bachelor's degree. On definitions, my point, as it relates to philosophy, is that you have to attend to the particular people you discuss things with to find common agreed to definitions. You dis'ed me for your fixed interpretation of the word "axiom" in your own head. To anyone versed in dialectic, you have to begin with definitions up front for the present argument. ...

 

This only reminds me of an elitist antique book seller I knew years ago who was so absurdly pedantic as to the specific pronunciation of "Principles of Mathematics" (Russel and Whitehead's three-volume tomb) when I inquired about whether he had it. I pronounce it as "prinsiples" just as we use it colloquially in my world and he insisted adamantly that I pronounce it "prinkiple" as if it had necessary significance. This is the same with your insistence on particular use and is why and where the image of many scientists get appropriately labeled as elitist (or snobbish). Who cares which words we use as long as when or where we use them we can relate. Are you not flexible enough to be able to interpret the sign "+" for meaning "inclusive OR" in logic because you only think of it as meaning "addition"?

Who's dissing who here?

[Scott Mayers]

 

 

With the greatest respect, perhaps you need to consider taking your own advice.

 

The (ancient) Greeks did indeed start with definitions.

But they then introduced a set of statements they called postulates and what we now call axioms.

They also introduced a second set of statements they called common notions and we also call axioms today.

 

All three of these types are statements made without proof.

The Greeks distinguised postulates as those special to the discipline under consideration and common notions as those with wider more general application beyond.

 

We call both of these axioms because we start with the most general and say that the common notions are induced from the more general.

 

What you have described above as postulates they called propositions.

Propositions are statements deduced from the original statements by some system of proof.

Today we call these theorems and lemmas.

Weird. I have a good stalked library of universally non-fiction material in all academic areas (a lot of which are text books too). I read across areas of different studies in contrast to the way you may have learned in more concentrated study. So I am well read and have no problem understanding the uses. I have and read Euclid's Elements as I have even my original high school texts on this. I've read across different algebra and calculus texts. I've read and studied logic in all its forms. The one common feature about all of them is that they use different terms in different areas that are not linked together. As such you learn to adapt to each understanding in practice.

 

Propositions are any "assumptions" traditionally used in Propositional logic (deals with any general human language without concern for external quantifiers.)

"Pro-" (= ahead of) "-posit-" (= to pose, as opposed to negating) a (posit)"-ion". The use is best in use for debate, politics, colloquial use, and logic.

 

Postulates are preferred uses of "assumptions" in modern geometry-related works or to reference self-evident propositions.

"Post-" as in command by an official posted edict without question up front. Also related to "pose" as well. This is used to beg the reader of the proof to intuit the reasoning without having to require more depth than what is self-evident.

 

Common Notions were the external assumptions of another trusted system of thought in common popular use. Like using popular observational conclusions common among people within geometry.

 

Axioms represented initially the conclusions within a system used as assumptions that were raised above a need for proof as they had already been proven within the system usually. They are also more common in use in pure numerical systems nowadays rather than geometry. But many use them distinctly to presume conclusions from a closely related. As akin to common notions, axioms can be the conclusions of another separate but related logic as borrowed in respect of it.

 

Theorems are the same as Axioms but within the system usually.

...

 

I don't need to go on as they are many. They all relate to "assumption" ("theorem" usually a conditional statement to some conclusion) in some way. It's not important.

 

"dis'ing" is short for "disrespecting". I purposely used it in contrast to the point I was making of how you are disrespecting the intention of the argument. If I'm doing this to you, at least you seem more informed than I am by my content rather than providing more depth to specify your meaning against me.

[Acme]

Weird. I have

a good stalked library of universally non-fiction material in all academic areas (a lot of which are text books too). I read across areas of different studies in contrast to the way you may have learned in more concentrated study. So I am well read and have no problem understanding the uses. I have and read Euclid's Elements as I have even my original high school texts on this. I've read across different algebra and calculus texts. I've read and studied logic in all its forms. The one common feature about all of them is that they use different terms in different areas that are not linked together. As such you learn to adapt to each understanding in practice.

 

Propositions are any "assumptions" traditionally used in Propositional logic (deals with any general human language without concern for external quantifiers.)

"Pro-" (= ahead of) "-posit-" (= to pose, as opposed to negating) a (posit)"-ion". The use is best in use for debate, politics, colloquial use, and logic.

 

Postulates are preferred uses of "assumptions" in modern geometry-related works or to reference self-evident propositions.

"Post-" as in command by an official posted edict without question up front. Also related to "pose" as well. This is used to beg the reader of the proof to intuit the reasoning without having to require more depth than what is self-evident.

 

Common Notions were the external assumptions of another trusted system of thought in common popular use. Like using popular observational conclusions common among people within geometry.

 

Axioms represented initially the conclusions within a system used as assumptions that were raised above a need for proof as they had already been proven within the system usually. They are also more common in use in pure numerical systems nowadays rather than geometry. But many use them distinctly to presume conclusions from a closely related. As akin to common notions, axioms can be the conclusions of another separate but related logic as borrowed in respect of it.

 

Theorems are the same as Axioms but within the system usually.

...

 

I don't need to go on as they are many. They all relate to "assumption" ("theorem" usually a conditional statement to some conclusion) in some way. It's not important.

 

"dis'ing" is short for "disrespecting". I purposely used it in contrast to the point I was making of how you are disrespecting the intention of the argument. If I'm doing this to you, at least you seem more informed than I am by my content rather than providing more depth to specify your meaning against me.

 

And no, I didn't buy the book. It was like $15, 000 as some collectible original. I opted to read the ones from my library.

 

...

This only reminds me of an elitist antique book seller I knew years ago who was so absurdly pedantic as to the specific yada yada yada...

This all just reminds me of those who have trolled here in the past using a mash of red herrings and other rhetorical verbiage as they let us know what bright beacons are they and what dim dullards are we. Boring.

[Scott Mayers]

This all just reminds me of those who have trolled here in the past using a mash of red herrings and other rhetorical verbiage as they let us know what bright beacons are they and what dim dullards are we. Boring.

Some, like you, started this against me here by your own feigned superiority and intentional treatment of me as some green thumb that I'm not. I am far from any troll and you'd see I don't attend to bother with those who do. But I come here as my namesake while you hide in obscure anonymity. So please, Acme, if you want to compete fairly, either qualify yourself with real credentials unanonymously or play fair and argue in context to the discussions. If you are demanding me of respect, you have to be just as willing to try too. Sorry for any reflection to you in kind. But you've got more to prove to me than I to you. I'm hoping devilsolution shows up to give me his response as I'm not interesting in discussing with others as it is clear you are only trying to 'put me in my place'.

 

I have nothing against you personally.

 

Thank you.

[Acme]

Some, like you, started this against me here by your own feigned superiority and intentional treatment of me as some green thumb that I'm not. I am far from any troll and you'd see I don't attend to bother with those who do. But I come here as my namesake while you hide in obscure anonymity. So please, Acme, if you want to compete fairly, either qualify yourself with real credentials unanonymously or play fair and argue in context to the discussions. If you are demanding me of respect, you have to be just as willing to try too. Sorry for any reflection to you in kind. But you've got more to prove to me than I to you. I'm hoping devilsolution shows up to give me his response as I'm not interesting in discussing with others as it is clear you are only trying to 'put me in my place'.

 

I have nothing against you personally.

 

Thank you.

Phhht. Anonymity is a red herring. We have no end of whining about it and here's just 3 examples:

Suffering from Online disinhibition effect

On posting etiquette and anonymity - Split from Why are Physics Speculations

Making your credentials known to new-comers

 

And in a breath you dis me for an authoritative tone while you bluster your own. Pot calls kettle black.

 

I created a similar topic regarding math and logic but my initial OP was totally incorrect and secondly this delves a little deeper into other factors of math.

...

One final note, it seems that maths is directly related to the physical world, we purposely use it for its functionality within the physical world. However it seems we have various aspects of the mathematic realm which has no bearing on physical reality (for example the use of dimensions in maths, which geometrically pile up, in comparison to our 3 dimensional world of x, y and z. Here the maths doesnt relate to reality). So is maths an extension of physics? Or is it the physical world that dictates the use or usefulness of math?

 

Regards and sorry for such the long post.

Math is sometimes a tool and sometimes a toy, though arguably toys are tools for play. Most of your post strikes me as better suited to a blog as it really doesn't foster much discussion. Since the questions you ask that I bolded are philosophical, we can expect argument ad infinitum with no resolution. What do you hope to achieve with this thread?

PS The earliest known example of math was for accounting and not physics.

1.2 Math at the Dawn of Time

[Scott Mayers]

Math is sometimes a tool and sometimes a toy, though arguably toys are tools for play. Most of your post strikes me as better suited to a blog as it really doesn't foster much discussion. Since the questions you ask that I bolded are philosophical, we can expect argument ad infinitum with no resolution. What do you hope to achieve with this thread?

I responded specifically to DevilSolution.

 

If math is only a 'tool' how can you trust it other than to how you predesign the tool to effectively meet your intended end goals, like an enzyme or catalyst that steps out unaffected? Are enzymes real? Thus, by 'tool' do you think that math or logic is unreal? Can you think of logic/math as empirically justified? If these are not real, then why not use other unreal tools like myths or religion to support your rationalizations?

[Acme]

I responded specifically to DevilSolution.

So?

 

If math is only a 'tool' how can you trust it other than to how you predesign the tool to effectively meet your intended end goals, ...

You wrongly presume all tools -mathematical or otherwise- are designed when in fact tools are also found -or discovered if you will- and used as-is. Moreover, adding 'only' misrepresents what I said and is quite unnecessary. And why the quote marks on 'tool'? Shall I give an accepted definition that fits my use of the term? No problem.

tool @ The Free Dictionary

4. (Tools) anything used as a means of performing an operation or achieving an end.

5. anything used as a means of accomplishing a task or purpose

 

... like an enzyme or catalyst that steps out unaffected?

Your writing is abtruse. 'Steps out unaffected'; what does that mean?

 

Are enzymes real? Thus, by 'tool' do you think that math or logic is unreal? Can you think of logic/math as empirically justified? If these are not real, then why not use other unreal tools like myths or religion to support your rationalizations?

Inasmuch as you wrongly presumed tools are designed and you didn't bother to consult a dictionary, your questions are non sequiturs.

[Scott Mayers]

A catalyst (enzyme in biochem) is a chemical that only acts in some reaction momentarily but itself does not change. That is, it steps in momentarily, participates in other reactions, but then steps out 'unaffected' or unchanged.

 

You assume the idea of using the tool doesn't have to be real in itself. If it is 'discovered' (I more than agree!), then is it not equally a function of nature and reality? Who says anything about 'design'? I'm more athiestic than you could likely imagine....nihilistic to top it off. I'm saying that logic or math act like the foundation and framework of a building at LEAST such that it allows one to place facts of reality in them with regards to science to find a means to connect them and draw conclusions. So it is not merely a catalyst, but requires to be substantial itself as it must remain in place to maintain support of the science. I'm for the empirical approach to logic too, ...a logical postivist one that got abandoned from science intentionally for expediency in politics. It's why you can't make Relativity fit with Quantum Mechanics for instance. They collide in severe contradiction. But we retain the interpretations intact of the initial author's 'story' in lieu of allowing better ones by reexamining the logic deductively. Today's science maintains a practice-only empiricism which bans logic other than as its political uses to prop up institutions and evade what the logical approach leads to.

 

I'm not against 'science'. Just the paradigm preserving a hands-off stance against philosophy and logic as its complementary function necessary to find real closure.

[Acme]

A catalyst (enzyme in biochem) is a chemical that only acts in some reaction momentarily but itself does not change. That is, it steps in momentarily, participates in other reactions, but then steps out 'unaffected' or unchanged.

OK...

You assume the idea of using the tool doesn't have to be real in itself.

I never assumed or implied any such thing.

 

Who says anything about 'design'?

Erhm...you when you wrote, "If math is only a 'tool' how can you trust it other than to how you predesign the tool ...".

 

I'm more athiestic than you could likely imagine....nihilistic to top it off. I'm saying that logic or math act like the foundation and framework of a building at LEAST such that it allows one to place facts of reality in them with regards to science to find a means to connect them and draw conclusions. So it is not merely a catalyst, but requires to be substantial itself as it must remain in place to maintain support of the science. I'm for the empirical approach to logic too, ...a logical postivist one that got abandoned from science intentionally for expediency in politics. It's why you can't make Relativity fit with Quantum Mechanics for instance. They collide in severe contradiction. But we retain the interpretations intact of the initial author's 'story' in lieu of allowing better ones by reexamining the logic deductively. Today's science maintains a practice-only empiricism which bans logic other than as its political uses to prop up institutions and evade what the logical approach leads to.

 

I'm not against 'science'. Just the paradigm preserving a hands-off stance against philosophy and logic as its complementary function necessary to find real closure.

That all sounds like personal issues outside of the scope of this thread.

I came into the discussion to say Gödel's theorems contradict your unsupported assertion

I believe there definitely IS an ultimate logic that encapsulates all other rationalizations in reality with respect to nature or totality itself.

Since you can't show you're right or that Gödel is wrong, this is where I leave it.

[Scott Mayers]

Gödel is wrong because he depended his argument on the idea that the abstract nature that defines the class of a universal to include itself. While I think this is alright too, in such an interpretation, he has no justice to assume that an infinite regress itself is not allowed.

 

A set that is defined as "the set of all sets" is one in which a member of it must also include this set too in an infinite regress. So his argument about such a possible set to be unable to be 'closed' or is "incomplete" falters because with respect to reality because he merely begs that the idea of infinity is not closed. But just because we as humans are locally unable to find closure with respect to totality as a whole, does not mean that totality itself finds this idea "incomplete".

 

For instance, the idea of a verb such as "walk" is infinite in this sense. Yet we still understand it finitely too. We do this when we create the term, "walking" to give it a noun meaning. [a Gerund]

 

The proof by Gödel was merely to show that all logical systems cannot be closed as he presented this type of argument to show that at least one will always exist that is in itself unclosed by his definition. However, we can also argue that the very process of him demonstrating the non-closure of all logics/maths can be closed in 'process' by redefining infinity in those logics generatively to become closed just as a verb can be translated into a noun. Even if another proof can come along to reinstate Gödel's theorem, another theorem can be presented generatively to prove it closed.

 

Another point is that if you accept Gödel's "Incompleteness Theorem" as is, this HAS to apply for all systems including science with it's stance on permanent tentativity in the same incompleteness. Most already default to assume that science does not nor cannot determine 'truth' because of this. Therefore, is science not just as rationally insignificant as logic in the same way? It is for this reason that I find science incomplete without the philosophy respecting logic as its underlying reality.

 

NOTE: You know that while Gödel's theorem is a theorem, what I present about redefining "infinity" can also disprove Gödel's theorem and act as a theorem itself. It makes his theorem itself without closure allowing for all logical systems.

Edited October 10, 2015 by Scott Mayers