Chess will never be solved, here's why

Well, of course some proofs may be not as convincing as I think, but it's undeniable that not only @tygxc have provided references, as everyone can see just reading previous posts. @tygxc thinks (and hopes) nobody ever read previous posts, so he repeats that no one has been able to find real problems in his ideas.

#3079

"I have provided proof and reference that about 10¹⁴ nodes have been searched to solve checkers, not 10⁷ like you repeated for many posts"
++ I originally wrote 10^9 as written in the first paper of Schaeffer. Indeed, his later paper said 10^7 nodes in the tree with each 10^7 nodes.

"I have also provided a paper that states clearly how the authors interpret van den Herik definition of "weak solution", while you didn't"
++ It is an interpretation. I still think 'any opposition' implies an act of opposing.

"MARattigan has showed the pathology in Stockfish."
++ He showed a pathology in KNN vs. KP which is a 5-men position known to be troublesome to engines and strongly solved. I invited him to provide an example for a relevant 7-men position like KRPP vs. KRP, but he has not found one so far.

"Your "evidence" is not a proof, otherwise chess would be already ultra-weakly solved."
++ personally I consider chess already ultra-weakly solved on the basis of the evidence I provided

"Just your word that you read the statement "provability is a higher degree of truth" on Scientific American proves nothing, sorry." ++ That is not my problem. I did not make it up and I read it there and then. If you do not believe that, then that is your problem.

"The real "accuracy" (not that computed by engines or by you) will be known (i.e. by proof) only after the solution."
++ Yes, that is true. It will all be known after it is done. The question now is if it is feasible. Likewise Schaeffer first wrote about checkers 10^9 nodes needed investigation and he ended up with 10^7 * 10^7

"If it was really known, the game would be already weakly solved."
++ Yes, that is true, but before somebody starts to do it, he needs to know if it is feasible.

"Again, you use "known" but not proven things to prove other things"
++ To weakly solve chess we must know if it is feasible. If we can only prove feasibility by solving chess, then it is chicken or egg. First the feasibility must be established even approximately.

#3081
"@tygxc thinks (and hopes) nobody ever read previous posts"
++ I hope people read, but I think people do not read or do not understand

"so he repeats that no one has been able to find real problems in his ideas."
++ I repeat hoping people read and understand.
The challenge is not to find problems, the challenge is to find solutions.

tygxc's postings remind me of arguments for a flat earth.

10⁷ is the top of the proof tree, not the nodes searched; 10¹⁴ are the nodes searched. You had even conceded that.

The special meaning of "act of opposing" (a limited subset of the legal moves that an opponent can play) that you think is correct, is not shared by game theorists. I have provided one reference specifically about van den Herik definition, because you use that, but other equivalent definitions are interpreted the same way. You, instead, did not provide any scientific paper supporting your personal interpretation (and common dictionaries are generally not sufficient for technical definitions). Using a personal interpretation of a definition is equivalent to use a personal definition. But that was not even the point. You said that only you provide references: that is clearly not true.

Exactly, personally, but no scientist announced that chess has been ultra-weakly solved. What is evident, is that your "evidence" is not a proof.

Again, you said you provided proofs and references. Previously, you said that we can use "known" but unproven things to prove something else and you based your claim on that statement, which is unreferenced; therefore it was a problem in your idea, not mine. But now you say that you think chess is ultra-weakly solved, so you don't even need that statement, right?

The feasibility of a weak solution is already proven. You want to prove the feasibility of your particular project, which is to get the solution in five years, but the problem is that your solution would be invalid. That's because your algorithm is supposed to cutoff lines on the basis of the "calculated" error rate per move, which is in turn based on the game-theoretic value, which is not proven to be a draw. If the solution "proved" that the game is indeed a draw, that "solution" would be a circular process: the game value is a draw, therefore the error rate is that, therefore those lines can be cut off, therefore... the game value is a draw. That is not acceptable as a proof, it's a fallacy. On the other hand, if the solution proved that the game value is not a draw, it would contradict the premise, and therefore the whole process would be inconsistent. Your "solution" might be feasible, but not valid, anyway. That's why no one attempted such a thing, it's not just a matter of money.

Nobody wasted money on false logic projects based on Sveshnikov saying something.
Figures.  Anybody who has that kind of money for big projects knows how to manage it much better than that.
That's why they've got it.
And so with @tygxc doing the pushing with the false logic -
well this isn't personal but refers to his posts -
that money is Safe !!

#3085

"10⁷ is the top of the proof tree, not the nodes searched; 10¹⁴ are the nodes searched."
++ Yes, I agree with that. My previous figure of 10^9 was from the elder paper by Schaeffer.

"The special meaning of "act of opposing" (a limited subset of the legal moves that an opponent can play) that you think is correct, is not shared by game theorists."
++ Van den Herik was/is one of the leading game theorists and approved using game knowledge to weakly solve a game as beneficial in his paper.

"known but unproven things"
If something is proven then it is true.
If something is not (yet) proven, then that does not mean it is not true.
There are things that are not yet proven but yet believed to be true based on available evidence.
Examples in mathematics are the Riemann Hypothesis and the Goldbach Conjecture and were Fermat's Last Theorem and the Four Color Theorem. Mathematicians work/worked on proving these, not on proving the contrary.

"The feasibility of a weak solution is already proven."
++ No, not formally. Strongly solving chess is possible, but not feasible. 10^44 nanoseconds and 10^44 bits of storage are not feasible with present technology. For weakly solving Sveshnikov said 5 years and I provided evidence that 5 years are plausible indeed.

"That's because your algorithm is supposed to cutoff lines on the basis of the "calculated" error rate per move"
++ Yes, to weakly solve chess it is essential to use game knowledge so as to reduce the number of nodes to be searched. 10^17 nodes is still a formidable number. Losing Chess needed 10^9 nodes, Checkers 10^14 nodes.

"On the other hand, if the solution proved that the game value is not a draw"
++ Nobody right in his mind believes chess to be a forced win. It would contradict all evidence.
A) AlphaZero autoplay with more draws with longer calculation time and even if stalemate is a win
B) ICCF WC with more draws in more recent years despite 7-men table base win claims > 50 moves without capture or pawn move allowed
C) Classical World Championships with nearly all draws in recent years
D) TCEC with high draw rate despite imposed unbalanced openings
E) Checkers (a draw) needing more nodes 10^14 than Losing Chess (a win) 10^9 despite being a simpler game.

"it's not just a matter of money."
++ It is a matter of money. 5 years to rent 3 cloud engines plus the assistants is a lot of money.

As usual, it is difficult to understand whether you do not understand what people write, or you just try to avoid issues.

I did not question the sequence of words van den Herik uses to define a weak solution, but the particular and personal meaning you give to those words. It is almost impossible that you misinterpreted me. As for the rest, some posts ago:

 Your citation is again out of the context. If you read carefully, In that section four methods are presented and they are all general methods, which can be applied to many games, not only chess. Those heuristics do not cut off lines arbitrarily. Alpha-beta pruning cuts off lines too, but when it is used to solve a game, that happens when the value of some positions has already been determined after a TBH (Table Base Hit) or equivalent (3-fold repetition, etc.); in that case and under some conditions, some lines can be indeed cut off, because it is proven that one or the other player would have a better path to choose. You, as usual, overgeneralize the idea.

Yes, but things believed to be true although not proven are conjectures. If they are used as premises, the result is a conjecture too, not a proof, therefore no solution.

We already know how to proceed, the problem is time. You aim to reduce time, but to do that you not only use conjectures to prove something else, which is already illogical; your method would use the conjectured game-theoretic value of chess to ultimately prove the game-theoretic value of chess (which is part of the solution): that is just a fallacious circular reasoning.

This is an argument from incredulity. We have not to believe anything, we have just to find out, but, for what above, your method would give only an approximation of a conjecture, at best.

I did not say it is not a matter of money. I said it is not just a matter of money, if a project to solve chess has not started yet (at least AFAIK).

Or both.  Three situations.
Apparently tygxc oscillates through the three.
And through others too. 
Often acting as if he doesn't understand what is being said to him while actually understanding some or all of it.
This would happen with 'flat earth' positions too.
Again not personal.  Simply addresses his postings.

#3088

"Next to brute-force methods it is often beneficial to incorporate knowledge-based methods in game-solving programs." ++ So I use knowledge to prune like 1 a4 and 1 e4 e5 Ba6.
It would be a waste of resources to look into something we know.

"Yes, but things believed to be true although not proven are conjectures."
++ Yes, formally they are hypotheses or conjectures.
However, there are hypotheses or conjectures with more evidence supporting them.

"If they are used as premises, the result is a conjecture too, not a proof, therefore no solution."
++ Assessing the feasibility of weakly solving chess need not be exact.

"that is just a fallacious circular reasoning."
++ No, that is done in various branches of science, e.g. mathematics and physics.
Example: What is the velocity v of an electron with charge e and mass m accelerated by a voltage V?
Solution: start by the hypothesis: v << c the velocity of light c. Thus use Newtonian mechanics:  eV = mv² / 2 and calculate v = sqrt (2eV / m). Now check if v << c then the hypothesis was true and the calulation valid, else change to relativistic mechanics.

"We have not to believe anything, we have just to find out"
++ In order to find out you have to start with a hypothesis.
E.g. if you try to prove the Riemann Hypothesis, then you start by believing it is true.

Do you even read what I wrote? You pick only what remotely can support your views, ignoring what can contradict them; it's a fallacy. After that statement you find in the paper what type of knowledge is used to find a solution. None of those heuristics cuts off lines arbitrarily the way you do. And you still use "know" to refer to conjectures, that are only believed true.

Whatever, but we need a proof, not a conjecture built upon conjectures.

You play with words. You are describing a process to get a solution, but that "solution" would not meet the criteria of a solution, which is a proof.

 Mmmmh... You maybe right that your assumption that the game-theoretic value is a draw does not lead to circular thinking, but the way you make use of it is fallacious, nonetheless. In the example you give, two different models can be used to answer the question. Both models are not proofs: they are theories, but one of them has been disproven (that is, it is not valid in all cases) the other one is general (so far). However, both models are not built on the premise that v has a particular value; that is, no particular value of v has been used to derive those models and they do not prevent the possibility to be falsified for some particular value of v, as indeed has happened for the classical model. Back to your method to find the solution, you use the conjectured value of the game as a basis to cut off lines, but those lines might actually falsify the conjecture. What I mean is that the conjecture may artificially support itself. If not circular reasoning, it's a form of faulty generalization.

No, why? An hypothesis is just an hypothesis. It may be true, it may be false. To seek objectivity, one should avoid the confirmation bias as much as possible.

#3091

"And you still use "know" to refer to conjectures"
++ "know" is chess knowledge and that is allowed and beneficial to weakly solve chess.
It is known that 1 e4 e5 2 Ba6 loses for white and that 1 a4 is not better than 1 e4 or 1 d4.

"Whatever, but we need a proof, not a conjecture built upon conjectures."
++ That is purism. It is allowed and beneficial to use chess knowledge to weakly solve chess.
It seems minds are changeing. First it was: "chess cannot be weakly solved as there are too many games/positions". Now it is: "no, you are not allowed to prune using chess knowledge."

"You are describing a process to get a solution, but that "solution" would not meet the criteria of a solution, which is a proof."
++ There are two different things:
1) assessing the feasibility, that need not be exact: approximate is enough
2) weakly solving chess. It needs to be exact and a proof, but use of chess knowledge is allowed and beneficial.

"You maybe right that your assumption that the game-theoretic value is a draw does not lead to circular thinking" ++ You begin too see

"but the way you make use of it is fallacious, nonetheless." ++ Fallacious because you say so?

"Both models are not proofs: they are theories, but one of them has been disproven"
++ Both models are knowledge. Both have been disproven. If V is very small, then quantum mechanics is needed. Relativistic quantum mechanics is rarely uses to calculate anything.
There are many more examples in science. E.g. the Maxwell equations are believed to be true. However they are too hard for most problems. In practice low frequency approximations, high frequency approximations, near field approximations, and far field approximations are used. Same methodology. First assume the approximation is valid. Then calculate with it. Then verify the assumed approximation is valid indeed.
The same with fluid mechanics. The Navier-Stokes equations are believed to be true, but are hard to use. Assume the Reynolds number is small. Calculate with the laminar flow hypothesis. Calculate the Reynolds number. If it is small, then the calculation is valid, else use another model. 

"you use the conjectured value of the game as a basis to cut off lines, but those lines might actually falsify the conjecture."
++ No, I do not use the game value. I use chess knowledge to cut of lines. I cut off 1 e4 e5 2 Ba6 because it is known to lose for white. I cut off 1 a4 because it is known not to be better than 1 e4 or 1 d4. I cut off some endgames with opposite colored bishops as these are known draws.

"What I mean is that the conjecture may artificially support itself."
++ Yes, my statement that 4 candidate moves suffice at 17 s/move on a 10^9 nodes/s engine for 1 error in 10^20 moves is based on the hypothesis that chess is a draw.
If that poses a fundamental problem to you, then there is another approach: successive approximations as also often used in other sciences.
First let the engine autoplay and verify it ends in a draw, or taken an ICCF WC drawn game.
Then in a 1st verification pass replace all white moves by the engine 1st alternative and check it is still a draw.
Then in a 2nd verification pass replace all white moves by the engine 2nd alternative and check it is still a draw.
Then in a 3rd verification pass replace all white moves by the engine 3rd alternative and check it is still a draw.
Add some more verification passes if you like.
Each verification pass increases the confidence level.
That is also common in many sciences. E.g. I have measured 1000 screws of a mass production of millions and I can say with 99.96% certainty that none exceeds the tolerance of 25 mm +- 0.1 mm.

"To seek objectivity, one should avoid the confirmation bias as much as possible."
++ There is subjectivity even in mathematics.
Trying to prove the Riemann Hypothesis is completely different from trying to disprove it.
It has nothing to do with bias, but with efficiency.
People trying to prove it are convinced by all failures to disprove it.

Chess "knowledge" is not proven, it's a bunch of conjectures "proven" by examples. For the third time, in the paragraph 5.2 of the paper you quoted, are described four algorithms: if you want to prove that that paragraph supports your claim, you should prove that any of those algorithms prunes 1. e4 e5 2. Ba6; otherwise, you are simply jumping to conclusions.

No. A solution is a proof, therefore it's simply not possible to call "solution" a conjecture, or anytihing based on a conjecture.

I never stated that chess cannot be solved because there are too many games/positions; you are confusing me with someone else.

A solution is a proof that a value v is the game-theoretic value of the game, and s is a strategy to achieve that value. The feasibility to determine s and v is already proven, because it can be done by exhaustive search; however, an exhaustive search would require too much time with current technology. Instead,
1) if you want to prove that a process P can determine s and v in 5 years, then you have to be as rigorous as for any other proof.
2) If you want to theorize that a process P can determine s and v in 5 years, then you have to use as few assumptions as possible and deduce the result rigorously.
3) If you want to theorize that a process P may determine s and v in 5 years, then it's basically worthless. It is a theory of a theory.
The third thing is what your process would accomplish, as explaind below.

Science provides a methodology, but it has to be clear that science does not prove anything. I think that Wikipedia is good enough to explain that:

   "Philosophers, such as Karl R. Popper, have provided influential theories of the scientific method within which scientific evidence plays a central role. In summary, Popper provides that a scientist creatively develops a theory that may be falsified by testing the theory against evidence or known facts. Popper's theory presents asimmetry in that evidence can prove a theory wrong, by establishing facts that are inconsistent with the theory. In contrast, evidence cannot prove a theory correct because other evidence, yet to be discovered, may exist that is inconsistent with the theory. [ . . . ] Alber Einstein said:

   The scientific theorist is not to be envied. For Nature, or more precisely experiment, is an inexorable and not very friendly judge of his work. It never says "Yes" to a theory. In the most favorable cases it says "Maybe", and in the great majority of cases simply "No". If an experiment agrees with a theory it means for the latter "Maybe", and if it does not agree it means "No". Probably every theory will someday experience "No" – most theories, soon after conception."[1]

The only real proof is a mathematical proof. It cannot be disproven if the axioms are not challenged. That's what a game-theoretic solution is. For chess, the axioms are the rules used to play the game.

Not only:

To everyone it is a problem in your theory. I also asked you to express your algorithm in pseudocode, which is more rigorous, instead of using common English. Just translate your statements in pseudocode or simple code with inputs and outputs, black blocks. Then we shall see how it works, even wihtout the details.

I don't know how you can prove how many attempts to prove or disprove the Riemann hypothesis have been made, but of course there is subjectivity in every choice. Nonetheless, a mathematical proof is objective, unless the axioms are challenged. You seem fixated with the Riemann hypothesis, but I was talking in general. Sometimes it is better to directly prove something, sometimes it is more efficient a reductio ad absurdum. This is independent of what one believes. Back to the topic, if you say: "1. e4 e5 2. Ba6 is likely to lose, let's make it the last candidate", it's ok; if you say: "1. e4 e5 2. Ba6 loses for sure, let's prune it immediately", it's not ok, because it is not proven (mathematically, I mean) that it loses.

Sveshnikov acknowledged that chess is an exact mathematical problem and I think he was aware that his idea of "solution" was not a game-theoretic solution, that's why in that interview he used the word "close" instead of "solve" and the word has been enclosed in scare quotes. If he meant a real weak solution he would have just stated that, don't you think?

#3093
Here is the google translation of the last part of the interview:
"What is your coaching credo?
I was able to teach master candidates to understand the opening, to write lectures on openings, materials for the encyclopedia. After all, you yourself wrote a lecture on the Spanish flute with g6, after that you read it to the first-class players and gave them a session in this opening with Black. You won with a score of 8:1 - you have delved into this position so much. And a year later you made a draw in this opening against Tal. And another student, with my help, wrote a lecture on the Panov attack in the Caro-Kann defense. The article is by far the best theoretical work on this option. At that time, my students did not yet use a computer, and only had what was available, plus my games.
The path that I myself walked, I wanted my students to pass. Now I think that the endgame should be studied in parallel with the opening. Chess is an exact mathematical problem, and you need to study it from two sides - the opening and the endgame. There is no such stage as the middle game. A well-studied opening is the middlegame. Often a deeply analyzed opening leads directly to the endgame. This is how Lev Polugaevsky understood chess. Of the great chess players, he is the closest to me. Soon all openings will be given exact scores, and if the variant is correct, it will lead to a technical endgame in which a draw will be achieved with accurate defense.
Today I teach children how to work with a computer. It is now impossible to study the opening without it. Give me five years, good assistants and the latest computers - I will bring all openings to technical endgames and "close" chess. I feel the strength in me.
Questioned by Eldar Mukhametov"

#3093

"Chess "knowledge" is not proven, it's a bunch of conjectures "proven" by examples."
++  We disagree on that. Some chess knowledge is absolute. See for example the scientific paper "Acquisition of Chess Knowledge in AlphaZero" it says knowledge, not conjectures.

"in the paragraph 5.2 of the paper you quoted, are described four algorithms"
++ Yes, the algorithm is brute force and making use of chess knowledge is said to be beneficial

"A solution is a proof" ++ Yes, but we disagree on what proof is acceptable.

"you are confusing me with someone else." ++ I am sorry for that.

"The feasibility to determine s and v is already proven, because it can be done by exhaustive search; however, an exhaustive search would require too much time with current technology."
++ Feasible means it can be done in a reasonable time, like 5 years. 5 million years is not feasible.

"1) if you want to prove that a process P can determine s and v in 5 years, then you have to be as rigorous as for any other proof." ++ No, about 5 years is enough. 5.24 or 4.89 is the same. It is proof of concept necessary as a prerequisite to embark on the solution.
"2) If you want to theorize that a process P can determine s and v in 5 years, then you have to use as few assumptions as possible and deduce the result rigorously." ++ Approximately is OK.
"3) If you want to theorize that a process P may determine s and v in 5 years, then it's basically worthless." ++ If the 'may' is plausible enough then that may be enough to start the solution.

"Science provides a methodology, but it has to be clear that science does not prove anything."
++ Science is a set of knowledge that is proven to satisfaction.

"I Popper provides that a scientist creatively develops a theory that may be falsified by testing the theory against evidence or known facts."
++ That is correct. I claim that evidence and known facts (expert opinions, TCEC, ICCF WC, AlphaZero autoplay, classical WC...) falsify the theories that chess is a forced win for white or for black. The are inconsistent with those theories. In your terms I use reductio ad absurdum to prove chess is not a win for white or for black. It is also what Sveshnikov says: "a draw will be achieved with accurate defense."

"The only real proof is a mathematical proof."
++ I disagree, it would mean that mathematics were the only science. Thermodynamics for example is taught now starting from axioms and using statistics. That is not how it has been developed. It used to be a practical way to predict the behavior of steam engines and the like.

"For chess, the axioms are the rules used to play the game."
++ That is right, but from these axioms Laws of Chess absolute chess knowledge has been deduced over centuries, in part by mathematicians like Anderssen, Steinitz, Lasker. Also Capablanca, Botvinnik and Sveshnikov had engineering backgrounds. So in weakly solving chess we can use that chess knowledge as beneficial. We are not obliged to re-invent all that.

"You seem fixated with the Riemann hypothesis"
++ I just take that as an iconic example of an unsolved mathematical problem.
Maybe the Goldbach Conjecture is a better example as it is simpler to state to the layman.

"This is independent of what one believes." ++ No, a mathematician starts by believing whether a conjecture is true or not and based on that he proceeds trying to prove or disprove it.

éif you say: "1. e4 e5 2. Ba6 loses for sure, let's prune it immediately", it's not ok, because it is not proven (mathematically, I mean) that it loses."
++ We diagree. In my opinion the grandmasters of the past have already proven that starting from the Laws of Chess and AlphaZero has only reconfirmed this knowledge. Likewise they have proven that 1 a4 is not superior to 1 e4 or 1 d4 and AlphaZero has reconfirmed this knowledge.

"in that interview he used the word "close" instead of "solve" and the word has been enclosed in scare quotes." ++ My Russian is not good enough. How is weakly solved in Russian? From the context it is clear he meant a real weak solution.

#3097
"They are obviously the words of a nutcase."
++ That says more about you than about Sveshnikov.
He had a master in engineering and was working on a PhD in engineering, while you dropped out in the first year.
He was a grandmaster and 65+ World Champion, while you are a weak player.
He was a professional chess analyst, you are not.

#3096
"van den Herik is a games theorist "
++ Yes, he is/was the most prominent authority on that subject.
You are neither a mathematician nor a game theoretician.
His definition is the definition. By the way it originates from Allen, the man who solved Connect Four.

Being good at gymnastics doesn't make someone an authority on biophysics.

That paper does not state that the knowledge is "absolute". AZ play is based on probabilities: it knows that playing 1. e4 e5 2. Nf3 gives (in its autoplay) a better expected outcome than playing 1. e4 e5 2. Ba6, but its conclusions are not definitive.

Those algorithms already use "knowledge" which is not really game-specific. They are less brute-force algorithms. After the sentence that you quoted so many times you can read: "The main advantage is providing an appropriate move ordering or selection in the search tree", not to prevent the exploration of some lines like you intend to do.

In my dictioinary, "feasible" means simply that it can be done (it derives from latin facere). That can be interpreted in different ways. Anyway, suffice that we do understand each other.

You say I am a purist? Of course 5 years can be little more or little less. The point is that if you want to prove that chess can be solved in such little time, your proof must be as rigorous any other proof.

If the result is not certain, it is abusive to call it a weak solution. A weak solution cannot be disproven, yours may be.

This statement is ambiguous, like too many other you use.

In your personal conception of what proofs and falsifications are.

No, it means that natural sciences can only diprove things, because they cannot prove them. Again from Wikipedia:

   "Proofs are examples of exhaustive deductive reasoning which establish logical certainity, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds true is not enough for a proof, which must demonstrate that the statement is true in all possible cases."[1]

So, since classical mechanics does not hold true on atomic scale or for speeds comparable with the speed of light c, it is disproven, even if it is obviously used for more common cases. That's why Einstein said that probably every theory will be disproven someday: they are just more or less inaccurate models.

I don't even know what you mean by "absolute knowledge". If you mean that it is proven that the strategic principles hold true in any position, you are for sure wrong, simply because we do not know all the possible cases. That's why a strategy has to be tested against all possible opponent's moves, to prove which is the minimum outcome that that strategy can forcingly achieve.

I don't even know where you take those type of informations. A mathematician can just start from axioms and/or theorems. Other times, they have just a question, not the possible answer, and use axioms and theorems to answer it, without even making hypotheses. Again, you are inclined to overgeneralize special cases.

In your opinion. See above.

I do not know Russian too, but I think he would have used too words, like in English. To me, it is not clear at all that he meant a real weak solution, especially because the word "close" is enclosed in scare quotes. Why use them?

I think it is good to keep the distinction between proofs and scientific reasoning clear, but for the proverbial "man in the street", it is convenient to lump them together as reliable ways to arrive at conclusions that can be reliably taken to be true thereafter.