My statement that you bolded does not imply either of those things, so I don't see the relevance in the response.A key aspect of precision involves objectivity and repeatability, which connect well to logic, whereas yes, accuracy relates to what I'm calling actual truth. Which returns us to the question of the connection between them.

I think I understand your querry now (maybe not). I go back to "a priori". If I understand you, Its a bit of a catch 22, if you allow as much.

I understand the ultimate truth to be only that which I can "know" thru my senses. . If I seek the answer to "Do you still exist after you leave the room and I can no longer eperience you thru my senses"?. Ultimate truth analysis is "I don't know". And so the truth as to whether you exists is based entirely upon my senses. Thats like a "real" a priori".

Now, that is the state of mind of a newborn. Then, Inotice that you ALWAYS return to the room return later. As I gain experience of the world thru my senses, I develope a logical reasoning that people don't stop existing just cause`I can't see them. So, I accept the limitations of my senses, in that, I will never know using my senses, if you exist when you are out of the room - and I develope a synthetic a priori. I imagine you outside of the room. This makes perfect sense according to my real a prioris, and so I adopt a synthetic a priori regarding your existence using LOGIC, not sensual ultimate truth. More?

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Where did everyone go?

There are subtle but important distinctions between formal logic and the way the word logic gets used in informal ways. Some people use "logic" to mean anything that seems reasonable. So if they know the Earth is spinning, and they know it would be hard to stop the Earth, or they know that the Sun has come up every morning for thousands of years of recorded history, they say it is "logical" that the Sun will come up tomorrow. But that's not quite what logic is, logic only played a small but clearly identifiable part in that process. Logic did not say the Earth is spinning, logic did not say it's hard to stop the Earth, and logic did not say that there is nothing around that is capable of stopping the Earth. Also, logic did not say that something that has happened every day for a long time should happen tomorrow. Those are all things that we say, those are all part of actual truth as judged and tested by us, not by logic. Those are patterns we have observed, part of our experience. Purely experiential truth, none of them are themselves examples of logic (though logic played a role in our assembling these experiential truths, just as grammar played a role in what I just wrote even though grammar has nothing to do with what I just wrote).

The statement 'logic did not say the earth is spinning' is counter-intuitive. From observation, astronomy has logically inferred that the earth is spinning with momentum. Hence it seems fair to say that logic does say the earth is spinning. Observation confirms its findings by logical processes to measure the regularity and predictability of claims. The discovery of the inner logic of the motion of the earth provides a conformity between our description and the actual truth. We now know for an actual truth that the earth is an oblate spheroid with predictable motion.

Logic does suggest the future should conform to the past. Synthetic a priori judgements, or necessary truths of logic, indicate that causality is as necessary as space and time as the condition of experience. As a matter of practical reason, we infer by logic that matter behaves according to scientific laws of cause and effect. Scientific laws have such strong empirical causal confirmation that they appear to be necessary truths of the operation of the universe. We infer causality by deduction from experience.

An example of a similar deductive process, from experience to universal truth, is in the way the helium-beryllium-carbon process was discovered to operate in stars. Hoyle observed there is a lot of carbon in the universe now, and deduced from this observation that the stellar process was the only way the universe could produce this carbon as we see it.

Hoyle's deductive method in this instance is an example of the broad principle that we observe cause and effect in operation, so deduce by logic that causality is intrinsic to the science of the universe, and have confidence the universal laws of science will continue to operate in the future.

This example of deductive reasoning indicates the flaw in the falsifiability theory of scientific knowledge as derived by Popper from Hume, that it fails to explain the relation between causality and truth. This argument is the nub of Kant's critique of Hume. Kant's explanation of the synthetic a priori necessary truths of reason shows that the positivist theory of scientific discovery accords too limited a role to reason and logic as the source of meaning and truth.

That is saying something different. Yes, astronomy used logic in making the case, but logic does not provide the content or the meaning of the case, that comes from experience/observations. Logic is only the mortar between the bricks. You don't build walls from mortar, you just can't build walls without mortar. If all integers are rational, and all rationals are real, then logic allows us to assert that all integers are real. Someone without the vaguest idea what a number is could check the logical validity of that syllogism, but would have no idea what it was saying or why it was important. Thus logic is not what tells us what it means for all integers to be real, because logic doesn't have the vaguest idea what an integer is or what a real is. Logic does not carry meaning, it only connects meaning.

That sounds like it is mixing in what I would call experiential truth with what is formal logic. We should reserve "logic" for the formal kind to avoid confusion, the formal kind allows no argument or difference of opinion. It seems to me the "inner logic" of the motion of the Earth does allow differences of opinion-- such as whether that "logic" involves Newtonian or Einsteinian language. There are different types of logic, but for the purposes of this thread we may imagine we have some standard in place. The motion of the Earth, on the other hand, can be described in many different ways even within a given standard theory (like, shall we say the orbit is an ellipse even though it isn't, shall we say it takes a certain amount of time to go around even though it doesn't, etc.).

I agree that is an actual truth, and note its "fuzzy" quality-- the Earth is not an oblate spheroid and its motion is not predictable (but I know what you mean, that's why it's an actual truth).
No, logic has no clue about what "should" be, it relies entirely on us to provide that kind of information, in the form of postulates and predicates.
To argue that cause and effect is a synthetic a priori truth, rather than simply an a posteriori experiential truth, you would need to be able to say that you can not imagine an effect without a cause, or a cause that led to no effects, or a cause that today leads to effect #1 but tomorrow leads to effect #2. Can you not imagine that? Could you claim that such a universe is impossible?

Again, it is experience that allows us to infer that, even though we may use logic as an aid to make the connections. What I'm calling actual truths are not devoid of the need for logical reasoning, but they are not the stuff of logic, because logic itself has no idea that these things are true. The sentences I write make sense because they follow rules of grammar, but a book on grammar would not give you any insight into what I'm saying if you were not constantly connecting my words to your own experiences (such as, the experience of learning what the words mean).
Yes, he connected a bunch of experiential truths via logic to anticipate a new experiential truth. This is very much an example of the complex interplay between logical truth and actual truth, but it does not say that the former is the latter. It just shows how important the connection is, yet that connection remains elusive and mysterious.
I agree that just what that role is is very much the issue here.

OK, I think we've sufficiently characterized the differences between truth and logic to address the grandaddy of logic results, the Godel proof that arithmetic is either incomplete (so there are actual arithmetic truths that are not provable from the axioms) or inconsistent (so there are actually untrue things that can be proved from the axioms). For those who don't know, that proof proceeds by translating the "Godel statement", which is essentially "this statement cannot be proven", into the language of arithmetic. Doing so generates an arithmetic statement that either cannot be proven, which would make it an actual truth that cannot be proven, or it can be proven, which would make it an actual untruth that can be proven. As no one knows if the arithmetic version of the Godel statement can be proven by formal logic or not, say from the Peano axioms, no one knows which is the case. Of course, we all tend to suspect that arithmetic is incomplete, because if it were inconsistent anything could be proven, including false things, and so far no one has ever found a proof for something false. But "so far" does not equal a logical truth, so we really just don't know for sure. Perhaps we can label it an actual truth, given the fuzziness and provisional character we allow ourselves when we apply that concept. The larger point is, one cannot understand the ramifications of the Godel proof, or even the proof itself, if one does not understand the difference between logical, symbolic, grammatical, bulletproof truth, and actual, meaningful, experiential, "fuzzy" truth.

The bottom line is, we now know that even in pure mathematics, there remains a difference of some kind, a crack (even if no more than a hairline fracture), between what is actually true and what is provably true.

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Logic is the grammar of truth.

Meaning and absolute certainty are incompatible, and profound meaning and absolute certainty are profoundly incompatible.

The only thing intelligence is capable of is recognizing itself.

At the risk of sounding like a whacko - I posed this same idea in a different thread. I presented it as "given a known radius, prove that a circle can be halfed".

I wanted to find out what the really good math people had to say about this because I was pondering that "if we still don't have a finite conclusion to pi, then there is no proveable 1/2 circle. By half I mean EXACTLY 1/2. If there is no exact whole, mathmatically, then there is no exact 1/2.

Without math, an exact 1/2 circle is true; with math it is not true or untrue.

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Where did everyone go?

I would say that math can tell you how to construct an exactly 1/2 circle, and prove that it is exactly 1/2 a circle, but you can't actually do it and get that exact 1/2 circle. So when you want to do something, and thus bring it into the realm of experiential truth, you have to accept the fuzzier character of any such "actual" truth, an actual 1/2 circle, which of course will never be the same as the 1/2 circle that math can prove. Note this isn't quite the same as the endpoint of the Godel proof-- in the case you are talking about, we have math proving a truth, and experience establishing a truth, that are not exactly the same but clearly quite closely related. In the Godel proof, we either have experience establishing a truth that mathematical logic is completely unable to assess, or we have experience establishing a truth that is the opposite of the outcome of the logical proof, and we just don't know which we have (and maybe never will). So it's worse than the disconnect of exact (syntactic) equality vs. near (meaningful) equality that we always face when comparing logical and actual truths, it's a more complete disconnect, but it only appears in very unusual and esoteric conditions. In other words, we see here two kinds of disconnects between logical and actual truth, one which is a tiny difference that pervades everything, and the other which is a big difference that shows up only quite rarely.

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Logic is the grammar of truth.

Meaning and absolute certainty are incompatible, and profound meaning and absolute certainty are profoundly incompatible.

The only thing intelligence is capable of is recognizing itself.

I mabe wrong, but it seems math can not tell me how to construct 1/2 circle:

Given diameter of 1, the circumference of a circle is pi. Show me the math that instructs me where to divide the circle in half, EXACTLY. Where do I divide? What is the value of each 1/2 circumference?

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Where did everyone go?

The construction may allow you to start anywhere you want, and divide the circle in half at that point. It would be similar to the construction for bisecting an angle with a compass. If the compass was perfect, and you could follow the directions perfectly, you'd exactly bisect the circle at whatever point you chose. The reality would be a nearly perfect bisection only, but the logical instructions are the same no matter how close to perfection is the reality.

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Where did everyone go?

Drawing an arbitary diameter of a given circle is constructible with compass and straight edge.

In fact, given the circle you only need the straight edge and knowledge of the the location of the center (which is required to construct he circle with the compass in the first place).

Math is just the technique for making logical proofs about sets of instructions. You seem to imagine that math is equations, but that is not the case, equations are only one of many techniques for proving things about instructions. When the instructions involve a component in the real world, like compasses and straight edges, then there is a real-world component to the process that is outside the mathematics, but the mathematics is still what you use to connect the instructions with what you want to accomplish. It is the way to prove that the instructions will work, and then only experience can test that they actually do. Ironically, the experience that does the testing is less exact than the math, which is why experience can only test the function of the math, not the math itself. There are a lot of potential misunderstandings about the connections between logic and truth, which this thread is about.

Is it assumed in straight edge and compass constructions that the logical puncture of the point of the compass remains available? You see, although I was swiftly shot down for my belief that Euclidean constructions and deductions can be made independent of their physical realisations, I still think that is true.

This question of whether causality is a truth of experience (a posteriori) or a truth of reason (a priori) seems to me an interesting one for your distinction between logical truth and actual truth. If causality is a necessary condition of experience, then we can't simply say we derive our knowledge of it from experience alone, but must consider the law of cause and effect as both a logical truth and an actual truth.

We observe the universe is matter in motion, operating by the causal processes formulated in the laws of physics. The 'actual truth' or 'what is really happening' is more complex in toto than we can perceive. However, actual truth has an inner logic which science can approach. For example Kepler's laws of motion describe a part of the inner logic of planetary motion, with strong predictive power.

By application of logical principles, including that the universe is self-consistent and that every effect has a cause, logic leads science to assume that causality is universal. A universe where causality was not consistent may have some interest as a thought experiment, but would seem different to the universe in which we live.

Science seeks to discover the inner logic of actual truth. By definition, logical truth must be consistent with actual truth.

I don't know what you mean by swiftly shot down, I think there is certainly a logical content to Euclidean "constructions", taken as mental constructions. To do them physically, one must enter the real world, with all its geometrical and measurement limitations. As it happens, these present no problems for Euclidean constructions, as the real world is spectacularly close to Euclid's conceptualizations, they just don't extend to the infinitely large or infinitely small.

I'm not sure why you say the law of cause and effect is a logical truth-- to me, it is the quintessential experiential truth. What's more, I've not seen that included in descriptions of logic, or axioms of mathematics. It seems like a postulate of physics, tested only in the real world but never strictly in the logical domain.
I agree, but this inner logic is itself an experiential truth, not a logical truth. You are talking about that mysterious connection between what logic can prove and what turns out to be true, and there is indeed a connection there, but logic can never prove that connection. I think the situation is quite analogous to how the meaning of our sentences here is supported by the use of correct grammar, but there is still no meaning in grammar.

That is a somewhat metaphorical use of "inner logic." I would say there is some inaccessible truth there, which we attempt to use our intellect and observation to construct a kind of facsimile or model of. The intellect provides the logic, the syntax, and the observations provide the words, the conjurers of experience, and together science writes its descriptions of what it can never completely characterize or reproduce. The role played by logic can be extracted as a specific piece of the process, the way we can extract the grammar from these sentences.
Science is not led to its postulates by logic, it is led there by observations. Yes, this is that mysterious connection between logic and discovered truth. But to what extent can we claim the universe is consistent? Consistent with what? Does it always obey our laws? No, it generally doesn't, so we must carefully control the circumstances of our experiments or we cannot predict the outcomes. That's why our laws take forms like "energy is conserved in a closed system." Does the universe provide us with any actual closed systems? No, it does not.

We already know that logical truth is not consistent with actual truth. I gave you the example of "this sentence is not a logical truth." That's an actual truth that is not a logical truth. It could instead be a logical truth that's not an actual truth, but it isn't expressed in the form of a logically assessible statement. Godel fixed that, using a system that we expect to be logically assessible (arithmetic), and that's why we don't know if arithmetic is incomplete or inconsistent but we do know it's one or the other.

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Logic is the grammar of truth.

Meaning and absolute certainty are incompatible, and profound meaning and absolute certainty are profoundly incompatible.

The only thing intelligence is capable of is recognizing itself.

That's the point and it reveals your error. I say that "logic is a process, truth is a result" but nowhere do I state that truth is the result of a logical process, which was your conclusion in your response. What was the source of your error, I do not know. Perhaps it was post hoc ergo propter hoc or perhaps you didn't realize that, "not all processes are logical" and "neither are all results truth", or ignored those corollaries.

I'm not sure how you arrive at precision involving "objectivity and repeatability" when the simpler explanation would seem to be the resolution of manipulation. The finer you can manipulate something the more precise its operation will be. The resolution of math is infinite, but the resolution of reality, especially human perception of it, is finite. Perhaps we invoke algebra as a mean of connecting them. Empirical data can be variable and using variables with numbers in mathematical calculations enables us to use logic with regards to reality.

With regards to halving a circle, it would seem to me to be quite easy as pi equals circumference divided by diameter and half of pi is half of that calculation: pi= (C/d)/2. While it might be more precise to attempt to describe it in numeric decimal notation, it is more accurate to describe it in symbolic fractional notation.

I don't think logic can connect meaning. It can merely perform operations. It is our perceptions that substitute meanings for operands where we wish.

BTW, can we maybe have the whole Godel Proof moved into another thread? Maybe this thread is supposed to be an attempt at the Socratic Method by Ken G, but I suspect more math people will see it and respond if they know that's what this thread was supposed to be about all along.

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"What you think you thought you saw you did not see." Agent J, MiB - Manhatten Bureau

Again I do not understand, I am not concluding that truth is the endpoint of a logical process, I'm defining "logical truth" as the endpoint of a logical process, with the additional caveat that the logical process is assumed to correspond to a consistent set of axioms (when the axioms are inconsistent, I would distinguish provability from logical truth). So I need draw no conclusions there, it is simply the definition of the words.
So far, you have not even identified an error, so you should start there before looking to explain its source.
Simple, resolution would have no meaning in the absence of repeatability.

Not if it isn't repeatable. Precise manipulation without repeatability is indistinguishable from chance.
With that I can certainly agree.
That is what connects meaning-- performing operations. Just look at the syntax in your own statements, you are using linguistic logic to connect meanings.
If anyone doesn't think the Godel proof is all about truth vs. logic, they don't know much about the Godel proof.

OK, did some reading, acquainted myself with Godel.

My first impression of the Godel Statement is that it actually streangthens the accuracy of arithmetic, in that:

Humans developed math to describe the world, however it is but a mere representation of the same. Humans experience the world thru their senses. If we wish to include in our world things that are not presently being sensed, we must start to experience the world thru a method beyond (besides) our senses and give equal reliance to that source as we do to our senses. However we can not prove or disprove things we come to "know" by this method using the same means we use to prove knowlege gained by our senses. Call this synthetic a priori. I can not touch it to prove it, but I believe it just the same.

No problem untill I want to share my version of reality with others. If I'm in a room with you and you say "prove that chair exists", I go pick it up and hand it to you then ask you to desribe it to me. When your done, I tell you "I would desribe it the exact same way". Proved. However, how do I go about proving my synthetic a prioris? Well, however I decide to so do, somewhere in the proof will be a true sentence I can't prove by "handing it to you". But, the sentence also happens to be one of your synthetic a prioris, and so you don't need proof, because you already know it as true.

Thus, unproveable truth is born. And since math is a representation of human understanding, and nothing more, and if math is to be as complete and consistant representation of human understanding as possible, and because science, thru its scientists, insist "descriptions of the word are only valid if supported by math" - then math itself will necisarily have to be based upon unproveable truths. Thus, accurate.

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Where did everyone go?

It may be worth remarking that we call it "Godel's Incompleteness Theorem" not "Godel's Inconsistency Theorem".

How so?

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Where did everyone go?

Because we do not deduce from Godel's result that arithmetic is inconsistent but that it is incomplete.

Nor did I. Am I missing something? I'm not a math guy.

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Where did everyone go?

My point was simply that people attach too much to the idea that Godel's result implies that mathematics is flawed.

Godel's original paper led to an enormous body of mathematics, including, for instance, Cohen's independence proofs in set theory which do, perhaps, have a bearing on the concept of truth within mathematics.

Oh. Then it appears we agree, as I percieve Godel's results as defining math as a more accurate platonic (taking librty to use as a noun).

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Where did everyone go?

I agree with much of what you are saying, but you are using a nonstandard meaning of "provable." In standard parlance, provability is exactly what mathematics has, and what showing people chairs (science) never has. That's because the descriptions of a chair must be framed in language, so even if their description is the same as yours, how do you prove they mean the same thing by the words? Meaning is only demonstrable, never provable.

One of the things accomplished by this thread is it gives us the tools to understand why what we call it is a misnomer. It is certainly not an incompleteness theorem. A certain mathematician on this forum has called me many unpleasant names for pointing out this fact, but it is a fact nevertheless. I can prove my claim quite easily-- if it were a theorem, then we could use it to prove other things, yes? For example, we could use it to prove the Godel statement, because the "theorem" requires that the Godel statement be true (can you arrive at it any other way?). In short, the status of the "theorem" is the same as the status of the Godel statement-- but the Godel statement asserts that it is not provable. Hence, if you think Godel proved an incompleteness theorem, you are saying that arithmetic is inconsistent. Put differently, there is no way to avoid the fact that logical truth and actual truth cannot be the same thing, even in mathematics, no matter how much we would like to pretend otherwise.

Hmm. Well my occasional posts here have been dismissed by both Ken G and Dr Rocket.

I don't think I have anything more to contribute, but I do suggest that people look at the extensive literature of Recursive Functions, Computability, and Unsolvability (much of it built on Godel's techniques and results).