[Author's note: I sometimes find it useful to repeat some of the basics of logic and rational thought. This is because it is essential to keep in mind that logic doesn't just happen, it requires a conscious effort to produce and test logical thoughts. This article is always available ot the accompanying website, atheism-analyzed.net.]
All rational thought and logic must, of necessity, be based upon something. It cannot be based on chaotic random principles, or there would never be an orderly thought procession. It cannot be based upon variable principles, or the logic would not be consistent. It cannot be based on principles obscured by complexity, or logic could not be understood. Logic must be – and is – based on principles of magnificent consistency, simplicity and clarity. Moreover, they are persistently universal.
Because of the consistency, logic works always and everywhere, within the confines of our universe. Because of the simplicity, logic builds directly from these basic principles without need for a still more basic set of principles to prop it up. And because of the clarity of these principles, logic can be easily understood without the confusion of complexity.
The FIRST PRINCIPLES are those that cannot be reduced any further to more basic concepts. They are fundamental. And they cannot be proven by any more basic principles, because there are none to use in the proof. So they are “seen” to be true by inspection; they are “obviously” true; they are “intuitively” true.
To restate that which was said before, without the fundamental “truths”, logic cannot exist. Logic must be based upon something simple, consistent, and clear. Yet there are those that reject the first principles. Rejecting them would have the effect of eliminating “rational” logic, and replacing it with a non-rational substitute. For example, Nietzsche rejected the first principles and developed an antirational philosophy. But any antirational philosophy must be considered fantasy by true rationalists.
Since rationalism depends entirely upon the validity of logic, it also then depends upon the validity of the first principles. This is especially true of empiricism, which depends on the principle of cause and effect, and the principle of non-contradiction. Were these not valid, empiricism would never have come into being.
So science, at least empirical science, is totally dependent upon the continuing validity and consistency across the universe of the first principles of logic and rational thought.
And so, science is based upon a set of unproven, and unprovable principles, that are known to be true only by intuition. Thus, if science is thought to be valid, then intuition is also assumed to be valid.
Last, if intuition is valid, then transcendence exists – because intuition is transcendent.
This all seemed like a fine and coherent article until your illogical conclusion at the end.
You stumbled across the "fallacy of generalizations", presuming that all "unproven and unprovable" principles are equally and non-cumulatively valid. So anything based on such principles is equivalent to everything else based on those principles.
I assert that that's not the case.
One, we don't know that the principle of cause and effect are "unprovable", only that it is unproven (but in fact, it has millenia of historical and circumstantial evidence that suggests that it is).
Two, it's arguable whether The principle of non-contradiction is indeed a "principle" or simply a rule of measure that allows us to make sense of the world. But there are plenty of things out there, such as quantum physics, that are accepted as true even though the measurements contradict each other. The fact that we perceive inconsistencies simply encourages rationalists to continue investigating the phenomena and try to find out what it is we're missing or not understanding.
Three, you haven't clearly defined what "principles," unproven or not, underlie intuition; but whatever they may be, it has been repeatedly shown that purely intuitive conclusions are often wrong and require additional empirical evidence to improve those odds. In other words, intuition has been shown to be a lousy source of truth, regardless what it's based upon.
Four, you are ignoring the evidence of history. Empiricism consistently comes out on top as a far more reliable indicator of truth than intuition. Yes, empiricism can still lead us to incorrect conclusions; but what's awesome about science (and what's lacking from intuition and faith) is that it has self-correcting mechanisms to winnow out false conclusions and correct them.
Humans have had intuition ever since we first walked out of the jungles of Africa. Yet the rules of empiricism and the scientific method, applied only over a short 300 years, have uncovered truths and created workable understandings of the world that have far surpassed anything humans "intuited" in the prior 200,000 years.
I also challenge your assertion that intuition is "transcendent". Most people would understand that intuition is simply a mental conclusion of the facts and experience that we have experienced. I don't know what's "transcendent" about that, except that that itself seems like an intuitive conclusion, which would be a circular argument.
080511 Skeptical of First Principles
”You stumbled across the "fallacy of generalizations", presuming that all "unproven and unprovable" principles are equally and non-cumulatively valid.”
This is not what I asserted. I asserted that certain unproven and unprovable principles are seen to be valid by self-evidence, and that self-evidence is available only to intuition, not to empirical experimental validation or falsification. This is not my own invention, it goes back to Aristotle and is stated in most text books on logic.
” So anything based on such principles is equivalent to everything else based on those principles.”
I don’t know what this means.
” One, we don't know that the principle of cause and effect are "unprovable", only that it is unproven (but in fact, it has millenia of historical and circumstantial evidence that suggests that it is).”
It suffers from the “inductive Fallacy” in that there is no possible proof of its validity tomorrow; hence it is unprovable.
” Two, it's arguable whether The principle of non-contradiction is indeed a "principle" or simply a rule of measure that allows us to make sense of the world. But there are plenty of things out there, such as quantum physics, that are accepted as true even though the measurements contradict each other. The fact that we perceive inconsistencies simply encourages rationalists to continue investigating the phenomena and try to find out what it is we're missing or not understanding.”
Quoting Quantum Physics without specific instances and studies is not an acceptable argument; that science is embryonic and changes almost daily. On the topic of pursuing contradictions as mere inconsistencies, why would one pursue the statement that A = !A? If rationalists do so, they are wasting their time.
” Three, you haven't clearly defined what "principles," unproven or not, underlie intuition; but whatever they may be, it has been repeatedly shown that purely intuitive conclusions are often wrong and require additional empirical evidence to improve those odds. In other words, intuition has been shown to be a lousy source of truth, regardless what it's based upon.”
Intuition is a human faculty of the human intellect. John Locke, “On Human Understanding”, Chapter: Reason, page 579:
“14. “Our highest degree of knowledge is intuitive, without reasoning. – Sone of the ideas that are in the mind , are so there that they can be by themselves immediately compared one with another: and in these the mind is able to perceive that they agree or disagree as clearly as that it has them. This the mind perceives that an arch of a circle is less than the whole circle, as clearly as it does the idea of a circle: and this therefore, as has been said, I call “intuitive knowledge”, which is certain beyond all doubt, and needs no probation, or can have any; this being the highest of all human certainty. In this consist s the evidence of those maxims which nobody has any doubt about, but everyman (does not, as is said, only assent to , but) knows to be true as soon as ever they are proposed to his understanding. In the discovery of and assent to these truths, there is no use of the discursive facility, no need of reasoning, but they are known by a superior and higher degree of evidence. “
(continued from above)
Now this from Bertrand Russell in “The Problems of Philosophy”, page 72, 73.
”In fact the principle is impossible to doubt, and its obviousness is so great that at first sight it seems almost trivial. Such principles, however, are not trivial to the philosopher, for they show that we may have indubitable knowledge which is in no way derived from objects of sense.
The above principle is merely one of a certain number of self-evident logical principles. Some at least of these principles must be granted before any argument or proof becomes possible. When some of them have been granted, others can be proved, though these others, so long as they are simple, are just as obvious as the principles taken for granted. For very good reason, thereof these principles have been singled out by tradition under the name of ‘Laws of Thought’.
They are as follows:
(1) The law of Identity: ‘Whatever is , is.”
(2) The law of contradiction: “Nothing can both be and not be”.
(3) The law of excluded Middle: “Everything must either be or not be.”
These three laws are samples of self-evident logical principles, but are not really more fundamental or more self-evident than various similar principles: for instance the one we considered just now, which states that what follows from a true premiss is true. The name, “Laws of Thought,” is also misleading , for what is important is not the fact that we think in accordance with these laws , but the fact that things behave in accordance with them; in other words, the fact that when we think in accordance with them we think truly.”
” Four, you are ignoring the evidence of history. Empiricism consistently comes out on top as a far more reliable indicator of truth than intuition. Yes, empiricism can still lead us to incorrect conclusions; but what's awesome about science (and what's lacking from intuition and faith) is that it has self-correcting mechanisms to winnow out false conclusions and correct them.”
Your scientism is duly noted, still self-evidence wins because science is based and dependent on the self-evidence of the First Principles stated, yet science is marred by both its innate inductive fallacy and associated deductive fallacy, which prevents science from ever producing “truth”. Science produces contingent factoids and is capable of no more than that.
(continued from above)
” Humans have had intuition ever since we first walked out of the jungles of Africa. Yet the rules of empiricism and the scientific method, applied only over a short 300 years, have uncovered truths and created workable understandings of the world that have far surpassed anything humans "intuited" in the prior 200,000 years.”
I think the above statements by Locke and Russell sufficiently disprove this.
” I also challenge your assertion that intuition is "transcendent". Most people would understand that intuition is simply a mental conclusion of the facts and experience that we have experienced. I don't know what's "transcendent" about that, except that that itself seems like an intuitive conclusion, which would be a circular argument.”
Intuitions are not material; they cannot be weighed nor measured, they assume no space or time or mass or energy. They cannot be removed from a person mechanically for dispassionate objective observations, nor can they be replicated by determining a cause and then forcing the result as an effect. So the intuition itself cannot be empirically falsified; only the material products resulting from the implementation of that which was intuited can be tested, and then only if the intuition concerned a material subject. If the subject of the intuition is metaphysical then there is no empirical recourse to verify or falsify even the subject of intuition. Skeptics might deny its validity, but they do so (as skeptics will) without adequate evidence to defend their rejection using the material evidence they require.
So I challenge your statement that “most people would understand…”, and why that, even if so, disproves any of the principles of self-evidence.
Just a quick question regarding your opinion of qm. Do you think Copenhagen qm scientists are contradicting themselves by denying that anything is causing the decay of radioactive elements on the basis that we can't do the experiments we need to because we alter the outcome of the experiment doing the experiment on the one hand, but on the other hand they're admitting that there is something for us to affect by doing the experiments?
Great article, even for atheists like me. I'm interested in the statement that intuition is transcendent, and that this proves other things are possible within the category of transcendence.
Is it possible that, in order to recognize that first principles are true or appropriate as the basis for logic, we need to first experience life so that we glean the principles by induction? After a certain number of patterns repeat, you come to the conclusion that there is some uniformity of nature, and can usefully be identified and contrasted with other patterns. Before you see examples by real experience, the concept of repeating patterns may or may not be intuitive.
I feel like induction and first principles are foundations of one another, rather than one of the other. I also find it hard to be certain that I have any intuition at all, although this may be a bit of a stretch to claim.
George Boole has shown that the First Principles derive solely from set theory without reference to physical objects. That means that induction is not necessary for "inducing" the First Principles from nature.
However, it can be argued that there is a circularity between any self-evidence and the prior existence of rational thought. This, however, is an acquisition issue for human minds, not an issue of prior existence in the universe. By that I mean this: these principles were applicable in the entire universe long before humans grasped them as universal fundamentals.
That humans came to understand those principles using induction, before understanding the principles surrounding the induction which they used, does not reduce the primacy of those first principles.
Intuition comes into play when the principles are tested, say by the use of Reductio Ad Absurdum. The chaos which would exist if the principles were false can be intuited, but not tested empirically.
For example, it would be extremely difficult to impossible to create a local container within which we could force the First Principles to NOT exist.
Even though there is no reason for these principles to exist when one considers the simplest of null states, the principles do exist and they form the basis for rational thinking, including induction, and deduction.
Further, it is not possible to deduce the non-existence of a rational source for these principles, from these principles, unless one exercises non-rational principles, specifically the infinite regress of universes. This infinite regress is both anti-rational and non-empirical and it also allows for the possibility of super sets of increasingly intelligent and powerful creatures, one of which might have created this universe, possibly as a subset of its own. From an empirical standpoint, this is purely magical thinking.
But within the confines of grounded, disciplined Aristotelian deduction, there is no infinite regression, no circular argument, no ungrounded argument which can be considered rational. And the grounding of rational deductive argument is always the First Principles. Arguments which violate those principles are not rational.
Finally, it can be argued that because humans cannot understand the first principles without first having attained rationality independently, that the first principles are trivial and in fact do not really exist since they cannot be proven. This is what Friedrich Nietzsche did in his book, Beyond Good and Evil, and he deduced that rationality did not exist either. Thus he developed his theory of Antirationality, and his conclusion that the only valid principle is the Will To Power. He was influential with Lenin, and was the official philosopher of the Third Reich.
The primary point to be made regarding transcendence, and what can be known about transcendent issues, is that it exists yet can be described using logic (which is itself transcendent, not being available to rocks, minerals, etc).
For example, the statement made above regarding the existence of First Principles. That First Principles exist, rather than the chaos expected from a perfectly simple null set, indicates a probable rational source (unless one invokes irrational "causation").
This probability cannot be calculated without the intuited sub-probabilities that are allowed in Bayesian hypothetical calculations. In fact, this probability is merely intuited, and is easily rejected under radical skepticism (another anti-rational thought process).
However, it is difficult to rationally support such denials. The concept that rationality, supported by First Principles, comes from a non-rational source is not a conclusion which is sustainable by rigorous logic; it is anti-rational and without grounding.
PS. The Boolean proof for the First Principles is here:
I think Nietzsche said the principle of equality can only apply to abstractions, and so it can only hold primacy over other abstractions, such as numbers or sets. It breaks down when applied to things we can actually experience, which are in a constant state of change, and where various limitations prevent us from proving they are equal, and which are subject to fabricated abstractions in order to make use of their "equality". This in turn leads it to wonder whether or not such principles require empirical verification, which undermines their primacy.
Or, put another way, I can only intuitively see that first principles apply to other intuitions, and fear there is a missing proof of application to all possible realities.
A principle of equality is not one of the Aristotelian principles of thought. Those are tautology; non-contradiction; excluded middle.
There are subsidiaries, including cause and effect, universality, etc. But I'm not familiar with "equality".
Also, these are "universals" which apply only to our universe; all possible realities would include total chaos and randomness within which no rationality could interpret that existence.
The First Principles are mathematically shown by Boole to apply to first order (binary) truths, and to fail to be comprehensible at higher orders. In other words, once Truth is asserted not to be absolute, chaos takes over.
The mathematics referenced above as proof of first principles used the notion of equality of things. Variables, which are abstractions of real things, are introduced and their equality compared. Without this, there is no proof. The variables, and, in fact, anything else one makes up or intuits, eventually must be applied to the observable world, either successfully or unsuccessfully. Otherwise, they are just proven as having applicability to other abstractions. This test is empirical and necessary for extension to reality.
Also, as to chaos: is that really a boolean state? Could we be capable of making up terminology by which truth applies universally to abstractions, but in reality applying those abstractions inevitably leads to some measure of unpredictability in reality. How would we know this is possible or not possible?
Actually, Boole's math does not reference "things" at all. It does reference equality of numbers in sets, and their icons, (all abstractions) under the principle of tautology (an abstract concept). And yes, taken to the limits of skepticism, this is circular, if one claims that the mathematics must have assumed first principles, in order to prove the Aristotelian first principles. That is where intuition comes in: the Aristotelian principles are first intuited to be self-obvious within our our experience of the universe, then mathematics is derived from them, and ultimately math returns to prove the first principles based in the further intuition of the validity of set theory.
Nietzsche, in "Beyond Good and Evil", merely denied that any principles could be intuited to be correct; thus he insisted that rational men would insist on empirical validation. But empiricism itself depends upon first principles, so his requirement is internally incoherent.
As for chaos and unknowabiliity, of course there is unknowability, as attested to by Godel, Heisenberg, philosophy of contingent science due to both falsifiability and the Inductive Fallacy, etc. Yet these do not prove the absence of a true base, but merely to our inability to have objective proof of it.
Even our intuition of the validity of the Aristotelian principles of valid thought is subject to the Induction Fallacy. But there are validating tests (Reductio) and there are principles of rational acceptability for dealing with Radical Skepticism.
Both the principle of noncontradiction and the principle of the excluded middle could be falsified. To my knowledge they have not been falsified throughout the millenia in which they have been defined.
Chaos might be defined as the set, [!order]. If one insists on the existence of infinite universes, ("many worlds" logic) then there must be one in which zero organizing principles exist. Our understanding of that would be "chaos".
The abstraction issue applies only to half of the utility of the first principles: ontology. The utility is found in every digital device, and even the "chaos" principle is found in electronics, for instance in flip-flops of the RS variety. If these devices receive proper boolean signals they perform per their truth tables. However, if they receive contradictory simultaneous signals, the device cannot resolve this to a truth table state, so the output signal wanders around randomly with no relationship to the inputs, unable to settle to a logic value. This is called "metastability" which is a logically undefinable, chaotic state, and it must be accounted for in digital designs in order to avoid chaotic operation.
Epistemology does not resolve to material issues, and remains in the realm of abstractions (truth, justice, intellect, and the principles of logic, themselves, for example).
This is an interesting topic, and I welcome this discussion.
I just realized that the principle of tautology could be considered a principle of equality. But it is not merely a principle of observed equality (ontlological, material equality), but also of non-material, epistemic equality and also defined equalities.
For tautology to be considered always referenced to material existence, a very rigid concept of Philosophical Materialism must be invoked, as a truth which is prior to the Aristotelian First Principles. If first principles must reference material existence, then Philosophical Materialism, as an a priori first principle, must also reference physical, material existence.
That means that Philosophical Materialism must be shown both valid and true using physical, material existence. That is both self-referential, and a Category Error; the sole existence of physical, material "things" cannot be demonstrated, much less proved, by referring to physical, material existence.
Further, Philosophical Materialism requires that epistemological concepts either be physical material "things", or not exist at all.
By asserting, without proof, that Philosophical Materialism is true, the concept of "truth" becomes chaotic, since there is no physical, material "thing" which can be identified as "truth". Truth has no material essence, so the declaration of the a priori primacy of Philosophical Materialism as a "true" first principle cannot be the case. The very concepts of both "truth" and Philosophical Materialism is destroyed by the assertion of PM as truth.
IF I understand Steve correctly,he's asserting that Math references and symbolizes physical objects only,which can be empirically detected.In other words,mass/energy presupposes mathematics.Mathematical truths are contingent on matter and therefore can't be considered to be neccessary and absolute truths because it is purely subjective and relative.What's mathematically true for some is false for others.There exists a possible world where 2+2=3,since is not conceptual nor transcendent,it must be found in nature,possibly under some rocks,inside photons,etc.
Do I understand you correctly thus far?
Hm. That would seem to suggest that, in a universe consisting of 95% dark mass and dark energy, a corresponding dark math, dark logic, and dark science, all of which leave no trace on the human senses or material sensors which humans develop. The characteristics of dark math, dark logic and dark science would be unknowable, yet they might dominate the universe in ways that are... dark... to humans.
Conceding the primacy of the "darks" (non-mass/energy) over the "visible" (mass/energy) would seem rational, then.
In other words, the non-material would dominate the material, given the state of cutting edge scientific knowledge.
Perhaps the Materialist needs an extra sense that would enable him to perceive Dark Matter/Energy.Since "6th sense" has already been associated with a transcendent intuitive ability,i think it's only fitting we label the extra materialist's perception (EMP) as the Dark Sense.This mysterious dark sense that allows the Materialist to perceive dark matter and energy whilst using his dark logic is believed to be located somewhere in the deep recesses of his brain.Albeit he cannot prove empirically nor syllogistically the existence of all things dark,he does however have his dark sense of humor to ridicule those who dare question his obscure superstitious beliefs.
On a more serious note,I really wish these guys could stick around longer.Atheists are fond of bailing just when things barely start heating up.Shame on them.
Yeah. They miss all the fun...
Re: "Math references and symbolizes physical objects only,which can be empirically detected":
I made a specific claim as to the proof above, that the variables were abstractions of reality. I made it in order to try to expose a contradiction. If they are not intended to apply to reality in any way, then they can only confirm other abstractions. Otherwise, if you always assume without inductive proof that abstractions really do apply to the real world, you are left with nonsense like negative numbers exist, rather than the more reasonable assertion that they plays a limited but incredibly useful role in mathematics and language.
In order to be a useful abstraction, math needs zero, multiplication and division for many of it's "proofs". Numbers we see in nature, such as the examples you gave, are universally consistent with division. Zero is not, and is granted an exception. It is absolutely critical for "proofs", though, such as the one above. Math is not even internally consistent.
a) Math is an incredibly useful but internally inconsistent abstraction which, if used to "prove" other abstractions, can provide contingent proof only.
b) The math abstraction is contingently consistent with the logical principles abstraction.
This is as far as it goes for me.
Re: 'Aristotelian principles are first intuited to be self-obvious within our our experience of the universe'
When I was first taught logic, my intuition was that it was an abstract tool for knowledge synthesis, and that people would construct syllogisms using logic, which sometimes are confirmed by induction, and in other cases are accepted without that verification requirement, such that:
Aristotelian principles are first intuited to be a good foundation for the abstract modelling of reality within our experience of the universe.
My intuition appears to contradict someone else's intuitions. How is this to be arbited?
”Zero is not, and is granted an exception. It is absolutely critical for "proofs", though, such as the one above. Math is not even internally consistent.”
The fact that division by zero is incomprehensible to mortal limitations does not actually make it internally inconsistent. The inability comprehend the nature of imaginary numbers (square root of -1), yet the ability to use them in describing the physical realm and even create physical systems indicates that they are not internally inconsistent. Math provides consistency.
”a) Math is an incredibly useful but internally inconsistent abstraction which, if used to "prove" other abstractions, can provide contingent proof only.
b) The math abstraction is contingently consistent with the logical principles abstraction.”
First, a) is internally contradictory; X cannot “provide contingent proof” if it really is internally inconsistent; also a) is contradicted by b) in the manner written. X cannot be “contingently consistent” if it really is internally inconsistent.
You have taken the liberty of violating the second law of thought, noncontradiction, while granting that law “contingent consistency”.
You seem to assert that X is only somewhat internally inconsistent. I suspect that you mean that there are mathematical features which are not comprehensible in the physical realm, therefore math is wrong in those instances. But it is not wrong; it provides consistent answers which are not physically realizable.
This violates your preconception. So which is wrong?
”This is as far as it goes for me.”
OK. You may place your limits wherever you choose; however, you have not disproved either the Aristotelian first principles, nor Boolean proof.
”When I was first taught logic, my intuition was that it was an abstract tool for knowledge synthesis, and that people would construct syllogisms using logic, which sometimes are confirmed by induction, and in other cases are accepted without that verification requirement, such that:
Aristotelian principles are first intuited to be a good foundation for the abstract modelling of reality within our experience of the universe.
My intuition appears to contradict someone else's intuitions. How is this to be arbited?”
You could prove that the Aristotelian principles are false, even contingently containing instances of absolute falseness. However, that would involve accepting a different grounding basis for your reasoning, and new principles for defining “true/false”. Since you intuit a non-Aristotelian grounding (or no grounding at all, a la Nietzsche), then your proof would be couched in terms other than deductive syllogisms, and your validation would be something other than Reductio Ad Absurdum. So your proof would need to start with some sort of support for your personal intuition of the new and different foundations of true and valid thought processing. Several millennia of logicians will have to be invalidated. Also, your all new grounding/validating process would need to account for the success of Aristotelian-grounded science and technology. Thus, you have your work cut out for you.
It is possible that you merely reject logic and have no alternative path to rational conclusions. That would render you “irrational” in the eyes of Aristotelian logicians and most of western societies. It appears, however, that you accept Artistotelianism but reserve the right not to accept it when you choose. That sort of position would appear “ornery” to most Aristotelians, I think.
Please keep me posted on the progress of your work should you pursue it, it should be interesting.
I would like to add emphasis to a point made above. Zero is a valid numerical concept.
How many Bandicoots are using your computer at this moment? Zero is an acceptable answer, and it is physically realizable by empirical inspection.
Again, it is not possible to conceptualize the quantity represented by a division by zero. And that does lead to other incomprehensible mathematical games when infinities are used. Yet, their physical incomprehensibility within the limits of human cognition does not render them invalid. It does render them not useful for calculations within the physical realm; there is no infinity within the known universe. This means that the physical limitations are the restriction point, not the math.
In fact, infinity is excluded from logic in the sense that infinite regressions are disallowed. Such regressions do not resolve to any knowable facts.
There is a current trend among cosmologists to allow infinities in their calculations, so long as they cancel. But that is also irrational under materialist restrictions, since unknowable A is not necessarily the same as unknowable B. Yet it might be. Infinity (divide by zero) is merely incomprehensible to material sensibilities, not obviously internally contradictory.
Another use for zero is in calculus: f(X)dX, where dX-> 0. Or, f(X,t) dX/dt, where dt ->0. Physically unrealizable, yet mathematically correct, useful, and internally consistent.
The same argument goes for imaginary numbers, which have no cognitive basis in materialist existence, yet are useful and give consistent results. Lack of material representation for the obviously useful notion of the square root of -1 falsifies the assertion that math, if it is not materially realizable, is internally inconsistent (contradictory). It obviously can be BOTH not materially realizable, AND internally consistent.
Were it truly internally contradictory (per the Aristotelian law of non-contradiction), then the results of its use would not be consistently correct. But they are. So intuition otherwise is not correct.
What has happened here is that the ideology of Philosophical Materialism has been smuggled into the description of mathematical constructs, specifically the divide by zero, and the square root of -1. Philosophical Materialism is an ideology which is itself internally contradictory, and which cannot – like any ideology – be applied to any first principles ad hoc. In fact, if any ideology is valid and true, it must be justified in a manner that is grounded in first principles, NOT used to justify or falsify first principles, in order to then justify itself.
” In order to be a useful abstraction, math needs zero, multiplication and division for many of it's "proofs". Numbers we see in nature, such as the examples you gave, are universally consistent with division. Zero is not, and is granted an exception. It is absolutely critical for "proofs", though, such as the one above. Math is not even internally consistent.”
1. Zero is not the exception: “nature” is the exception. In other words, materialist concepts of the totality of existence are granted the exception, in order to maintain a semblance of internal consistency – in an ideology which is internally contradictory.
2. The reason zero “is absolutely critical for "proofs" is because mathematics is incomplete without it. That it produces results which are materially incomprehensible is a limitation of material existence, and a falsification of Philosophical Materialism.
3. Math is definitely internally consistent, even when the results cannot be interpreted by reference to physical “things”. The issue is not with math, it is with Materialism.
Alright, I promise to stop after this.
You have internalized an intuition of Philosophical Materialism as a fundamental first principle of existence. All other principles, including Aristotelian first principles, and math, are subordinate to Philosophical Materialism.
The presupposition that there is no existence beyond the visible, tangible is completely without any hope of validation. It cannot be a principle of physics, or of inductive/deductive empirical observation. It is not an objective fact. It is purely a constriction placed on reality by virtue of personal preference and desire.
The non-existence of non-material existence is therefore merely an ideological principle. There is no logical reason to accept it, outside of personal proclivity.
Thus it cannot be a priori to Aristotelian first principles or mathematics.
Very nicely responded. I'll only post again here if I can contribute something without being repetitive.
Thanks for the conversation, it has been interesting. Comment at will.
I said (paraphrasing you):Math references and symbolizes physical objects only,which can be empirically detected":
You responded:I made a specific claim as to the proof above, that the variables were abstractions of reality.
One,your response agrees with my paraphrasing.In Atheistic terms,reality is always material,therefore subjected to empiricism.
Two,you are incorrect regarding the nature of variables because they can by definition represent any element or value with or without referencing physical objects.
"My intuition appears to contradict someone else's intuitions. How is this to be arbited?"
Which axiomatic beliefs does your intuition contradict?
Also,I'm not aware of any contradictions in math.The closest thing in math that seems circular,in my understanding,is that set theory is both the foundation and a branch of math.Somewhat self-referential but not neccessarily inconsistent.
The premiere "paradox" in set theory was discovered by Russell/Whitehead as they attempted to derive a unified theory of mathematics. The "set of all sets" must contain itself, but cannot. The idea of a unification of math was pursued by Kurt Godel who demonstrated the unknowability of the existence of all true statements in a given system, thus requiring a superset for validation of the unified set, a superset which is not part of the original set. But obviously the superset would also require its own superset II, etc., ad infinitum... unless, of course, there is an authoritative reason to stop the run-away infinite progression. Under theism that reason would be the creating intellect of the creating agent. Under Philosophical Materialism, the infinite progression is necessary.
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