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I believe that science must rest on an a priori foundation (and is not empirical) and I wanted to use this forum to get a sense for how people might respond to these ideas. Here are the reasons I give for why science must ultimately have an a priori foundation:

 

1. Mathematics is a priori. I don't believe this can seriously be doubted.

Mathematical knowledge is universal (i.e. 2+2 always =4) and sense experience cannot provide universal knowledge (per the problem of induction). 

Mathematical formalism (the idea that math is "analytic" or just a matter of the extrapolation of concepts whose definitions we create) was defeated by Kurt Godel's incompleteness proof. 

2. The problem of induction basically shows that science either has an a priori foundation, or cannot constitute anything like knowledge. (All the universal statements in which science consists cannot be known by experience.  No matter how many white people I meet who are thieves, I am never justified in concluding "all white people are thieves."  Likewise, no matter how many times I find that a reaction has an equal and opposite reaction, I am never justified in concluding that "for every action there is an equal and opposite reaction."  Etc. 

3. Many scientific discoveries (including many of the most important ones in history) take place by means of thinking. "Thought experiments" are a major means of making scientific discoveries. Thus Newton and Einstein made their major discoveries just "by thinking about it for a long time" (rather than by experiments or observations). 

4. Many scientific claims clearly go far beyond what could be justified by experiment. To use a simple example, Newton's 3rd law is a universal statement about every action. But neither Newton nor anyone else since has ever tested every action in the history of the universe. To draw universal laws from a handful of examples would be a gross fallacy if science were empirical. 

5. Simple mechanics is clearly intuited a priori. Levers, wheels, and gears all work in ways that can be clearly intuited. Ironically, what happens to pool balls in Hume's classic example is also intuited a priori. I believe much (but not all) of classical physics is also based on a priori intuition. 

6. I believe that experiments and the scientific method interact with a priori intuitions in the following way. From a known class of events (by events I mean a physical phenomenon that can be repeatedly tested), scientists use a sort of repeated-hypothetical-deduction process to come up with a "hypothesis" (a general statement) that fits with the set of known events, eliminate those that lead to contradictions, and when they find one that does not contradict any known events, design experiments to test whether other events entailed by the hypothesis actually obtain. 

 

Thoughts?

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@John

 

"Mathematics is a priori. I don't believe this can seriously be doubted."  "Mathematical knowledge is universal"

 

 

Really?  Did the North American Indians, in 1,200 AD, have a Universal "a priori" knowledge of Newton's Fluxions, or Leibinz's Infinitesimals?  Or the modern-day idea of limits, which changed calculus in the 20th Century?  And which may be revised, yet again, in the future?

 

If mathematics is "a priori", then why do we constantly revise it?  Why are there significant, virulent disagreements regarding mathematical foundationalism among mathematicians?

 

The whole concept of "a priori" knowledge is meaningless, spooky supernaturalism drivel.

 

Thoughts?

Edited by New Buddha
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John M,

 

When you say that Newton and Einstein made their discoveries by thinking for a long time, you forget that they were thinking about physical measurements. For example, Einstein thought about how while moving under the influence of gravity alone, you do not feel accelerated but rather weightless. He reasoned that if you do not feel accelerated, then you are in fact not accelerated. Even so, gravitational paths are curved and so there must be a theory of gravity where gravitational paths are unaccelerated and yet curved. It is a natural outome from Riemannian geometry that on curved spaces there are unaccelerated paths that are curved. It was natural then for Einstein to suppose that the curving of spacetime is caused by masses and the simplest idea is that there is a direction proportion between spacetime curvature and the mass-energy-momentum of spacetime. This leads quickly to the general theory of relativity. So when you say that these men arrived at their ideas by pure thought, that is not strictly speaking true. Their thoughts reflected observable facts. As a case in point, after Einstein derived his general theory, he tested it on a well-known problem with Mercury's orbit and found that his gravitational theory exactly accounted for the observational discrepancies that had been a conundrum for sixty years. This was not abstract thinking disconnected from reality. It was thinking founded in observable reality.

 

You say, "Mathematical formalism (the idea that math is "analytic" or just a matter of the extrapolation of concepts whose definitions we create) was defeated by Kurt Godel's incompleteness proof." It seems that Godel's Incompleteness Theorem implies quite the opposite of what you are saying. For example, Godel proved that you cannot prove the Continuum Hypothesis. Paul Cohen proved that you cannot disprove the Continuum Hypothesis. It follows that this hypothesis is beyond logic and represents pure rationalism since it is also beyond observation.

 

There is a lot wrong with your proposed theory, but this will do for now.

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John, have you read Dr.Peikoff's article on the analytic-synthetic dichotomy?

Also you should know that there are Oist who are familiar with epistemology enough to realize that "space-time" is not a substance and that any theory that presupposes this is philosophically corrupt nonsense. See David Harriman's lecture 4 from The Philosophical Corruption of Physics.

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If mathematics is "a priori", then why do we constantly revise it?  Why are there significant, virulent disagreements regarding mathematical foundationalism among mathematicians?

I think you misunderstand "a priori" here. It isn't referring to inborn knowledge. It is referring to what you can figure out from logic or purely abstractions, without reference to the real world. So any disagreements or revisions are a matter of correcting logic, or coming up with new types of abstractions separate from observations in the world. "A priori" is still spooky, but your arguments don't apply.

So I'll give brief comments on John's 6 points.

1. Sense experience can and does provide the basis to know what '2' and '4' mean, even what '+' means. To know addition, you need to know what a single entity is and about what happens when you combine them. The incompleteness theorem doesn't deny that math concepts can and are built from other concepts, it only denies that a system of math can prove itself with its own methods.

2. You are only talking about fallibility here. For Objectivism, knowledge is not about universality, and induction is about gathering information about existents, not forming propositions that apply at all times in all contexts. ALL knowledge for Objectivist epistemology is contextual.

3. Their thought experiments are based on knowledge and facts from experience. The thought experiment doesn't "materialize" from the abyss.

4. This is a standard argument of skepticism, Hume style. The best answer I can give to this briefly is to refer to 2 again.

5. I don't get how simple mechanics are "clearly" intuited a priori. People easily come up with answers to simple mechanics, and maybe intuitively, but why is that necessarily a priori?

6. I wouldn't use the term "a priori" here at all, but I think I agree with the gist of this point.

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1.  Mathematics is derived from data that has been codified into  principles, which in mathematics is called theorems and such (sorry, been while). Simply put, man did not learn 1+1=2 because it was revealed through some revelation or built in his DNA then build from there. He observed it then drew conclusion, which goes for most of science outside mystic/modern methods that do just that (Example - God is given so I go looking for prof).
 
2. There is no problem with induction.  
 
As for your justification, it simply shows someone using induction poorly while dropping context on other facts that they know, for example human free will in the given example.  Failing to use all facts available means they failed to induct properly.  User error, not system error.  In fact, this is why I think most professors don't like Integration - It requires them to use their mind and be open to the fact that they may make a mistake and have to start again. 
 
3. Thought experiments work to a point of offering creative thoughts but they still have to be observed then data drawn from observation and honestly tested against all known knowledge - See point number 2.  When you don't do this you get ID or cats who can be alive and dead at the same time, and other such nonsense. 
 
4.  To require every lead known and unknown answered is to demand omniscience and omnipotence.  Or to put it simply - Because we are not God we cannot learn through observation.  The amount of data required to induct is determined by you based on your knowledge known and data required to make the logical connection.  It also requires you to understand that it is within all data known and can be modified later when more knowledge is know.  
 
Example: Observation shows that water boils at 100c.  I do not need to observe this a million time in every conceivable way to prove it.  It happens repeatedly within x time and I can reasonable confirm that water boils at 100c.  Later on someone tells me they were visiting friends on Mount McKinley and it boiled at a lower temperature.  After a lot of investigation I learn that water boil at different temperatures relative to sea level.  Is my original point wrong?  No - It is still right, but now I know more about water.  Scientists, after a lot of observation and data collecting resolve how boiling temp changes relative to sea level then come up with a means of tracking this for all heights relative to sea level versus the substance (which I believe is known as a bar).  
 
Moral of the story - Induction requires one to think and then it requires one to be honest and update as needed when more information becomes available. 
 
5.  Really - I learned how to use a lever or the wheel instinctively like a bolt from heaven?  No observation or trial and error needed?  COME ON. 
 
6.  This just a reiteration for the thought experiment under number 3 above.  
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I believe that science must rest on an a priori foundation (and is not empirical) and I wanted to use this forum to get a sense for how people might respond to these ideas. Here are the reasons I give for why science must ultimately have an a priori foundation:

 

1. Mathematics is a priori. I don't believe this can seriously be doubted.

Mathematical knowledge is universal (i.e. 2+2 always =4) and sense experience cannot provide universal knowledge (per the problem of induction). 

Mathematical formalism (the idea that math is "analytic" or just a matter of the extrapolation of concepts whose definitions we create) was defeated by Kurt Godel's incompleteness proof. 

2. The problem of induction basically shows that science either has an a priori foundation, or cannot constitute anything like knowledge. (All the universal statements in which science consists cannot be known by experience.  No matter how many white people I meet who are thieves, I am never justified in concluding "all white people are thieves."  Likewise, no matter how many times I find that a reaction has an equal and opposite reaction, I am never justified in concluding that "for every action there is an equal and opposite reaction."  Etc. 

3. Many scientific discoveries (including many of the most important ones in history) take place by means of thinking. "Thought experiments" are a major means of making scientific discoveries. Thus Newton and Einstein made their major discoveries just "by thinking about it for a long time" (rather than by experiments or observations). 

4. Many scientific claims clearly go far beyond what could be justified by experiment. To use a simple example, Newton's 3rd law is a universal statement about every action. But neither Newton nor anyone else since has ever tested every action in the history of the universe. To draw universal laws from a handful of examples would be a gross fallacy if science were empirical. 

5. Simple mechanics is clearly intuited a priori. Levers, wheels, and gears all work in ways that can be clearly intuited. Ironically, what happens to pool balls in Hume's classic example is also intuited a priori. I believe much (but not all) of classical physics is also based on a priori intuition. 

6. I believe that experiments and the scientific method interact with a priori intuitions in the following way. From a known class of events (by events I mean a physical phenomenon that can be repeatedly tested), scientists use a sort of repeated-hypothetical-deduction process to come up with a "hypothesis" (a general statement) that fits with the set of known events, eliminate those that lead to contradictions, and when they find one that does not contradict any known events, design experiments to test whether other events entailed by the hypothesis actually obtain. 

 

Thoughts?

 

In the Kantian sense, a priori simply refers to mental categories. For him, these  were space and time. 

Science and math are possible because we innately possess said capacities.

 

In as much as post-Kantians have claimed to have discovered more specific innate capacities (geometry, heuristic vs analytical thought, etc), the counterpush begins with citing Einstein's general relativity: space and time are real, objective, and mind-independent.

 

Beyond that, you seem to be saying that our ability to form intuitions is innate, as well. Moreover, that our capacity to intuit throws a wrench into the wheel of induction.

 

To both, i would answer, "yes, indeed", furthermore adding that educated guessing forms a great deal of the scientific process, too. Outside of my own domain of literature, i'm an abductive sort of gurl.

 

Andie

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Thanks everyone!

 

Maybe I should clear define what I mean by a priori.  I mean something that we "just know" and which can neither be provided, nor justified, by sense experience.  If you ask me how we know it, my answer is that I don't know. 

 

Buddha, the ability to learn and come to know in mathematics (and science) is consistent with that.  There hasn't been major changes in mathematics, but there were certainly whole societies that had not discovered it.  This is obviously because it takes a good deal of sustained thought to do so. 

 

By the definition above, I think there's no need to classify a priori as "spooky supernatural drivel."  It could be because of the way our brains are, because of the way that the universe is, or it could be something supernatural.  But it doesn't have to be.  Chomsky is a good naturalist who is widely held to have proven a priori concepts in linguistics. 

 

Plasmatic, I haven't read either of those.  Where can I find them?  I'm not understanding your point about space-time not being a substance or its relevance. 

 

Eiuol, you're right about how I would define a priori.  Sense experience does cause it to arise, but it goes beyond what sense experience could provide.  For example, when I learn about the properties of triangles, I come to know certain things about how triangle must always be--which constitutes universal knowledge (which sense experience could not provide).  Granted that someone drew a "triangle" on the blackboard to help me "see" it, but (a) the triangle that he drew was not a real triangle (subject to small imperfections, etc.), and (B) it caused me to see in my minds' eye what I, in a sense, already knew (like Socrates showing that the slave boy knows the pythagorean theorem). 

 

For the sake of clarity I'll try to classify my responses under the topics of my original points (but give them names so we don't have to keep scrolling up to re-read them). 

 

Math (#1) - Eiuol, I don't think sense experience gives us the "idea of a single entity."  It only gives us the experience of seeing a constantly changing arrangement of shapes and colors, sounds, etc.  There are no single entities in our sense experience except insofar as abstract them.  But to do so, you need to understand the concepts by which you could do so.  Sense experience could never give you more than its own content, which plainly does not include the concept of unity (or any concepts at all--it is just itself!).  Sort of like the difference between a video camera that can record something, and a computer that can interpret that "sense experience."  If the CPU doesn't ALREADY have the concepts programmed in it to do the interpretation, it could never get those from its "sense experience" by being hooked up to a camera. 

 

Spiral, if man's observations of 1+1 = 2 is based on observation, then how are we justified in drawing the universal conclusion that 1+1 will ALWAYS equal two?  (Just because something has occurred a certain way in the past or in one case, or even a million, we are not justified in concluding that it ALWAYS will be so.)

 

Godel (#1a) - Aleph, Godel's Incompleteness Theorem proves precisely what I'm saying.  Google "Godel's incompleteness theorem formalism."  There is near consensus that Godel's theorem brought an end to formalism, the idea that mathematics was purely the extrapolation of man-made concepts. 

 

Induction (#2) - Eiuol, fair enough, but then science cannot constitute knowledge.  The scientific method commits the fallacy of assuming the consequent, and thus is no better than prejudice or "jumping to conclusions."  But I don't think that's true. 

 

Spiral, there is a problem of induction.  It's not clear what you're proposing is the solution to it, but at least you'll have to explain it if you claim that practically the most important problem in philosophy has  been solved!

 

Thought Experiments (#3) - Aleph, you're right, but less so with Newton.  But remember that something being a priori (as I tried to define it above) does not preclude the fact that it arose in response to sense experience.  The reason that the distinction is important is because it goes beyond what could be provided from sense experience alone!

 

Eiuol, no, thought experiments don't materialize from the abyss, but no more can they be justified by sense experience alone!  Better to say that the concepts that underlie science don't materialize from the abyss of sense experience!

 

Spiral, not only is it not clear to me how you're connecting apriorism to Schoedinger's cat, but it IS clear to me how there is a direct line from empiricism --> logical positivism --> empircal criterion of meaning --> Heisenberg's uncertainty principle --> shroedinger's cat.

 

Scientific Conclusions Beyond Sense Experience (#4) - Euili, I'm not advocating skepticism--but Humean arguments do present a choice between skepticism and rationalism (in my view). 

 

Spiral--you say, "induction requires one to think"--yes, but thinking itself is not something that can be provided by sense experience.  A mind is not a blank slate insofar as a slate cannot think!  And one should start by thinking about the problem of induction! 

 

Simple Mechanics (#5) - Eiuol, intuitive is a synonym for a priori (by my meaning). 

 

Spiral, yes.  Simply draw a lever on a paper and you can intuit what will happen without bothering to build it.  It doesn't have to be seen as a "bolt from heaven," and the "bolt from heaven or sense experience" dichotomy is, to me, very loose thinking. 

 

Hypothesis Generation (#6) - Euiol, fair enough.  But the thinking that underlays the scientific method relies on a number of concepts which can (in my mind) never be justified or taken from sense experience, and out of which actual universal knowledge (such as that of mathematics) can be extrapolated (I'm thinking things like math, logic, cause-and-effect, etc.).  That's what I mean by a priori. 

 

Spiral, yes, that's probably true. 

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JM,

 

You go too far when you say, "There is near consensus that Godel's theorem brought an end to formalism, the idea that mathematics was purely the extrapolation of man-made concepts." Rather, Godel showed that mathematics contains theorems that cannot be proved through mathematical logic, and that mathematics cannot be used to prove its own consistency. Neither of those mean that mathematics is not purely man-made. Yes, Godel's work demonstrated that Hilbert's second problem of finding a complete consistent set of axioms for mathematics was impossible. That was the end of Hilbert's formalism. That does not mean that mathematics was not a human creation. The parallel postulate and the continuum hypothesis were clearly human creations. While the parallel postulate at least has some basis in perception, the continuum hypothesis certainly does not. You must admit that mathematics contains concepts that are pure human creations disconnected from any perception. Just as there are forms of mathematics where the parallel postulate is untrue, there are forms of mathematics where the continuum hypothesis is untrue. You are free to choose which form of mathematics you wish to adopt. Certainly that freedom implies that these principles are human-made.

 

Concerning Newton, we know that his inverse-square theory of gravitation was first postulated by Bullialdus in 1645, and Bullialdus reasoned by analogy with light for which there existed empirical data. Newton had the advantage of calculus by which he could demonstrate that the inverse-square theory implied Kepler's Laws. Kepler's Laws were arrived at using Tycho Brahe's data and so were resolutely rooted in observational data and reams of computations. From Brahe's data, Kepler must have realized that the sun was not at the center of Mars' orbit and so Copernicus was not completely correct. He must also have realized that Mars' orbit was not circular, and in searching for an alternative to circular orbits his familiarity with conic sections gave him an obvious alternative to test. None of this would have occurred to someone who was not completely familiar with Brahe's data concerning Mars.

 

A priori knowledge is akin to Aristotle's understanding of science. He postulated that objects have an inherent velocity. In addition, he taught that in order to arrive at correct physical theories that it was unnecessary to appeal to physical experiment since pure reason was sufficient. Galileo proved him wrong. In advocating "a priori" knowledge, you are advocating a regression to pre-Galilean, Aristotelian rationalism that would set us back into the dark ages.

 

There are within Objectivism certain concepts that are taken a priori, such as logic and causality. Mathematics is not a priori in the sense you suggest.

Edited by aleph_1
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Simple Mechanics (#5) - Eiuol, intuitive is a synonym for a priori (by my meaning). 

This part I think is important and I'd like a more detailed answer before replying to the rest. How are simple mechanics intuitive and aren't anything at all to do with experience? How does the intuition arise? To me, I'd say the intuitions (and errors) towards simple mechanics involves a lot of experience!

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Spiral, if man's observations of 1+1 = 2 is based on observation, then how are we justified in drawing the universal conclusion that 1+1 will ALWAYS equal two?  (Just because something has occurred a certain way in the past or in one case, or even a million, we are not justified in concluding that it ALWAYS will be so.)

 

I already answered that in my statement on knowledge does not require you to be God(s).  I also answered this on my statement on the nature of integration in that it is contextual and when to make the logical conclusion.  You build data until you make the logic leap based on the data, what you need to integrate, and delimited by your current knowledge. 

 

If you see one apple, and then see another apple, and together they make two, I would hope you can draw a conclusion that 1+1=2 pretty fast after several observations.   Honestly, if you need to see two objects a million times with multiple scenarios and STILL can’t figure it out, then demand a revelation so you can grasp it, then there is a bigger issue with the user. 

 

 

 

 

Spiral, not only is it not clear to me how you're connecting apriorism to Schoedinger's cat, but it IS clear to me how there is a direct line from empiricism --> logical positivism --> empircal criterion of meaning --> Heisenberg's uncertainty principle --> shroedinger's cat.

 

I have no idea what this means, honestly.  Sorry.  I was just making fun of my favorite example of bad analogies.  That is the worse example I can think of a scientist what happens who project an idea (i.e. you’re a priori) then sets to demonstrate it without stopping to ask himself if it even makes sense. 

 

 

 

 

Spiral--you say, "induction requires one to think"--yes, but thinking itself is not something that can be provided by sense experience.  A mind is not a blank slate insofar as a slate cannot think!  And one should start by thinking about the problem of induction!

 

Thinking, or consciousness, simply is.  Its content is not.  We are born tablarosa and from there we have to build the content of what we know.  How we use it is also a matter of free will.  I suspect it’s that fact that scare scientist away from integration. 

 

 

 

Spiral, yes.  Simply draw a lever on a paper and you can intuit what will happen without bothering to build it.  It doesn't have to be seen as a "bolt from heaven," and the "bolt from heaven or sense experience" dichotomy is, to me, very loose thinking. 

 

See – We agree, even if it is a primitive example by your esteem, that we do learn from observation.  Building common ground J

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This is separate since it is a longer explanation... 

 

Spiral, there is a problem of induction.  It's not clear what you're proposing is the solution to it, but at least you'll have to explain it if you claim that practically the most important problem in philosophy has  been solved!

 

There is no problem.  Yes – Professors say so but that is to be expected from institutes of "higher" learning.   And there has been a debate, some good and some ridiculous (see hume) as well, bu the answer is there if you go looking.  I didn't make this up but read it myself.  

 

The classical problem of Integration is when you can go from data observation to a logical conclusion.  This is the “logic leap”.  The solution is the demands of “integration” and “delimitation”. 

 

Integration is determined by the nature of the subject and ease of observation.  You need little observation to figure out 1+1=2.  You will need more to figure you 12x12=144.  You will need truck loads to discover Calculous.

 

Delimitation is the body of your existing knowledge that will affect the time needed.  Or to put it another way, what you know and how it relates to what you’re learning.  If something contradicts what you know, then you know you’re in trouble with your current logical conclusion (thus my knock against the cat analogy). 

 

Back to the math example, you have two people looking at the same data and they are working to discover that 12x12=144.  The first person has no knowledge of math and struggles a long time to put the data together and discover the fact (if ever). 

 

The second person has been taught addition and subtraction already.  This person has a body of knowledge that will delimit the time to draw the conclusion.  Further, it expands his ability to grasp it versus if he started from scratch.  His time needed to make the logic leap was delimited by his existing knowledge. 

 

Incidentally, this is what makes language and reason so powerful for man.  Not only can he integrate advance concepts from observation but he can relay that to others and thus we build a body of knowledge that lets us continue to grow instead of reinventing the wheel every generation.

 

I hope that helps. 

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John said:

 

Plasmatic, I haven't read either of those. Where can I find them? I'm not understanding your point about space-time not being a substance or its relevance.

 

 

 

 

The article on the analytic-synthetic dichotomy is in Introduction to Objectivist Epistemology. There is work to be done on this topic as relatest to induction in my view and Professor McCaskey is making great inroads on this topic :

 

http://www.johnmccaskey.com/joomla/index.php/blog/79-analytic

 

 

 

 

The lecture can be found here:

https://estore.aynrand.org/p/387/the-philosophic-corruption-of-physics-mp3-download

 

My comment about space time was related to your being confronted with a Objectivist touting the substantival interpretation of "space-time". My intent is to give a counter example of another Oist rejecting this nonsense based on the foundational nature of a normative epistemology. For detailed explanition of the factual, historical philosophicl basis of such pjysics see the lecture. Andie Holland's posts are a prime example of all the philosophical garbage that preceded the quantum mysticism of the new physics. That is, what Andie is selling is what Harriman calls the "quantum fairytale" that physicist were forced reluctantly to renounce foundationalism, induction, realism, identity and causality for a statistical-mathematical description of appearences.

 

https://www.youtube.com/watch?v=aow8hVpdSHQ

 

It was philosophy that lead to the current state of physics and not alleged experiments. The postmodernist myths that are being peddled by Andie are to be fought on philosophical grounds and until Oist wake up they will be pawns in propagation of this fairytale "narrative"....The primary obstacle here is ignorance of the philosophical history of the actors in the drama that unfolded in physics..

 

 

Jon said:

 

Maybe I should clear define what I mean by a priori. I mean something that we "just know" and which can neither be provided, nor justified, by sense experience. If you ask me how we know it, my answer is that I don't know.

 

Yes, Its clear that your notion of the "apriori" is unlike the neutered interpretation imputed to you. This is a straightforward assertion of innate knowledge. Mrs Rand gives a detailed explaination of how axioms and first-level concepts-philosophical primaries are abstracted from perceptual experience in ITOE. I reccomend you do your own homework because you are likely to get 5 different answer on this forum. A good way to mitigate such varied interpretations here would be to request that whoever claims to be representing Oist premises provide quotes-references.

 

John said:

By the definition above, I think there's no need to classify a priori as "spooky supernatural drivel." It could be because of the way our brains are, because of the way that the universe is, or it could be something supernatural. But it doesn't have to be. Chomsky is a good naturalist who is widely held to have proven a priori concepts in linguistics.

 

John Searle gives an exellent criticism of Chomsky by pointing to Dan Everrette's work with the language of the Piraha tribe.

 

John said:

 

Spiral, if man's observations of 1+1 = 2 is based on observation, then how are we justified in drawing the universal conclusion that 1+1 will ALWAYS equal two? (Just because something has occurred a certain way in the past or in one case, or even a million, we are not justified in concluding that it ALWAYS will be so.)

 

 

This is where understanding the linguistic nature of concepts come in. There are no non-conceptual languages and likewise no non linguistic concepts. The abillity to use language properly is part of conceptualization. The fact is, the question "how are we justified in drawing the universal conclusion that 1+1 will ALWAYS equal two?" is answered by knowing what in experience the symbols you are referring to mean.

 

That is, the statement "1+1 may not always equal 2" is a nonsensical missuse of language. That is, it is a contradiction. Logic is the method of preserving meaning via linguistic-conceptual means. The meaning of a concept is its referents. The contradiction of philosophic primaries is known- understood by reducing the symbolic-linguistic-conceptual tools to their origin in experience and thereby identifying the conditions that satisfy and establish the correspondence of the statement.

 

John said:

 

Godel (#1a) - Aleph, Godel's Incompleteness Theorem proves precisely what I'm saying. Google "Godel's incompleteness theorem formalism." There is near consensus that Godel's theorem brought an end to formalism, the idea that mathematics was purely the extrapolation of man-made concepts.

 

Consensus is not a truth value. Do you relate the self reflexive nature of Godel.s theorom to the type of tautalogical circularity of axioms? That is, in faundationalist premises there is at bottom a type of justification that is non-propositional and the irreducible base on which "proof" rests. There is no meta context to retreat to because concepts are integrations of sense data. Consequently the axiom of existence is the widest and irreducible context at the foundation of knowledge which:

 

" [...] is a fact which cannot be analyzed (i.e., broken into components) or derived from antecedent facts."

http://aynrandlexicon.com/lexicon/irreducible_primaries.html

 

There is no "outside" system to justify sense data.

Edited by Plasmatic
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I said:

 

Consensus is not a truth value. Do you relate the self reflexive nature of Godel.s theorom to the type of tautalogical circularity of axioms? That is, in faundationalist premises there is at bottom a type of justification that is non-propositional and the irreducible base on which "proof" rests. There is no meta context to retreat to because concepts are integrations of sense data. Consequently the axiom of existence is the widest and irreducible context at the foundation of knowledge which:

 

It occured to me that it can be interpreted that I I impute the non-propositional type of foundationalism of Oism to foundationalism as such. I should have said:

 

"That is, in the Oist type of foundationalism there is at the bottom of all knowledge a type of justification that is non-propositional which serves as the irreducible base on which "proof" rests

Edited by Plasmatic
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Aleph, my understanding of the much-touted curved space in which parallel lines meet, is that a line in curved space curves with that space, and so is really not a line, and so the parallel postulate is in no way disproved by non-Euclidian geometry, as is often alleged.  You can't choose which form of mathematics to adopt.  There is only one--the real one.  And the existence of certain riddles and controversies in a very few areas of mathematics does not prove we can simply "choose which math to adopt," and that on the basis of that freedom, conclude that it's man-made.  There are right and wrong answers in mathematics (except, again, in the handful of areas where there remains mystery or controversy).  Even in those, there is likely a right answer even if we don't know it yet (there was a time when much of math was not yet discovered).  And I maintain what I said about what Godel proved meaning that mathematics was not man-made (which is to say, "merely axiomatic" or formal).  That's how he himself interpreted it and that interpretation is widely accepted in those who have not invested too much publically in the opposite ideas to consider following the argument to change their minds. 

 

Bullialdus's hypothesis (of which I was not aware until your post) actually seems to me to be an excellent case in point.  There's no way he had the tools to measure the intensity of light in 1650!  It follows from the idea of something's intensity being spread out over a greater area that it would occur over the surface of a sphere.  If you trace all the points a certain distance from something, you have a sphere--and if you "spread out" the force of that something over the surface of that sphere, you arrive at the inverse square law. 

 

Eiuol, I agree that we are made aware of our intuitions by experience.  But they go beyond what sense experience alone can provide.  The sense in which they are intuitive is that I can draw a simple machine on a piece of paper, show it to a child he will be able to tell just from looking at the paper and thinking about it, not only what will happen when I build that machine, but what MUST happen!  Sense experience can never tell us what must always happen.  It can just tell us what we've sensed. 

 

Spiral, I don't believe you understand the problem of induction.  "If you see one apple, and then another (etc.)," you say, "I hope you can draw the conclusion pretty fast!"  But if you meet one Dutch person who is tall, and then another, and then another, at what point are you justified in drawing the conclusion "Dutch people are always tall," in the same way you can draw the conclusion, "1+1=2"?   Experience + induction can NEVER lead to a universal statement (such as those of mathematics and science).  I think the ambiguity in your explanation in contained in the way you're using the phrase, "logical leap," and this will be clear if you try to define it precisely. 

 

Plasmatic, in reference to the article you cited about the analytic-synthetic distinction.  The author says,

 

"definitions that are formed with reference to things in the world are called a posteriori. (They are posterior to experience.) Those that could be formed without reference to the way the world is would be a priori. (They are prior to experience.) But concepts can’t be formed without reference to the way the world is."

 

I don't accept that way of defining a priori.  Rather than the distinction being about what is referenced by the concept, it is about what justifies them.  While a posteriori statements are justified by sense experience alone,  a priori statements are justified independent of experience.  The author himself admits the following:

 

"Such mature essentialized definitions are the building blocks of the exact sciences. They make it possible to have truly universal statements, statements that allow no exception. If what you measure doesn’t come out to be the ratio of voltage to current, then what you are measuring is not resistance. Simple. If the angles don’t add up to 180°, then the figure isn’t a planar triangle, because that sum can be derived from the very definition of a planar triangle."

 

That is pretty much what I'm saying in this post.  But such statements can never be justified by pure experience, per the problem of induction.  And they are NOT analytic--that is precisley what Kurt Godel proved they are not (at least with mathematics)! 

 

I haven't had time to listen to the lecture yet, but I agree with what you're saying about the state of physics being deplorable.  It is precisely empiricism that has lead to the current situation in physics. 

 

What is ITOE?  (I'm assuming the last two words are "Objectivist Epistemology…"?)

 

The description you give about "where understanding the linguistic nature of concepts come in," seems to be precisely what Godel proved was not the case (that mathematics was a matter of the extrapolation of man-made concepts).  Logic is what you can use FROM 1+1=2.  But I am talking about how you get there in the first place.  Neither logic alone, nor experience alone, nor the two working together can get you there!

 

You say "the meaning of a concept is its referents."  What is the referent of the number 1?

 

You lost me at the bottom of your post. 

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Aleph, my understanding of the much-touted curved space in which parallel lines meet, is that a line in curved space curves with that space, and so is really not a line, and so the parallel postulate is in no way disproved by non-Euclidian geometry, as is often alleged.  You can't choose which form of mathematics to adopt.  There is only one--the real one.  And the existence of certain riddles and controversies in a very few areas of mathematics does not prove we can simply "choose which math to adopt," and that on the basis of that freedom, conclude that it's man-made.  There are right and wrong answers in mathematics (except, again, in the handful of areas where there remains mystery or controversy).  Even in those, there is likely a right answer even if we don't know it yet (there was a time when much of math was not yet discovered).  And I maintain what I said about what Godel proved meaning that mathematics was not man-made (which is to say, "merely axiomatic" or formal).  That's how he himself interpreted it and that interpretation is widely accepted in those who have not invested too much publically in the opposite ideas to consider following the argument to change their minds. 

 

Bullialdus's hypothesis (of which I was not aware until your post) actually seems to me to be an excellent case in point.  There's no way he had the tools to measure the intensity of light in 1650!  It follows from the idea of something's intensity being spread out over a greater area that it would occur over the surface of a sphere.  If you trace all the points a certain distance from something, you have a sphere--and if you "spread out" the force of that something over the surface of that sphere, you arrive at the inverse square law. 

 

Eiuol, I agree that we are made aware of our intuitions by experience.  But they go beyond what sense experience alone can provide.  The sense in which they are intuitive is that I can draw a simple machine on a piece of paper, show it to a child he will be able to tell just from looking at the paper and thinking about it, not only what will happen when I build that machine, but what MUST happen!  Sense experience can never tell us what must always happen.  It can just tell us what we've sensed. 

 

Spiral, I don't believe you understand the problem of induction.  "If you see one apple, and then another (etc.)," you say, "I hope you can draw the conclusion pretty fast!"  But if you meet one Dutch person who is tall, and then another, and then another, at what point are you justified in drawing the conclusion "Dutch people are always tall," in the same way you can draw the conclusion, "1+1=2"?   Experience + induction can NEVER lead to a universal statement (such as those of mathematics and science).  I think the ambiguity in your explanation in contained in the way you're using the phrase, "logical leap," and this will be clear if you try to define it precisely. 

 

Plasmatic, in reference to the article you cited about the analytic-synthetic distinction.  The author says,

 

"definitions that are formed with reference to things in the world are called a posteriori. (They are posterior to experience.) Those that could be formed without reference to the way the world is would be a priori. (They are prior to experience.) But concepts can’t be formed without reference to the way the world is."

 

I don't accept that way of defining a priori.  Rather than the distinction being about what is referenced by the concept, it is about what justifies them.  While a posteriori statements are justified by sense experience alone,  a priori statements are justified independent of experience.  The author himself admits the following:

 

"Such mature essentialized definitions are the building blocks of the exact sciences. They make it possible to have truly universal statements, statements that allow no exception. If what you measure doesn’t come out to be the ratio of voltage to current, then what you are measuring is not resistance. Simple. If the angles don’t add up to 180°, then the figure isn’t a planar triangle, because that sum can be derived from the very definition of a planar triangle."

 

That is pretty much what I'm saying in this post.  But such statements can never be justified by pure experience, per the problem of induction.  And they are NOT analytic--that is precisley what Kurt Godel proved they are not (at least with mathematics)! 

 

I haven't had time to listen to the lecture yet, but I agree with what you're saying about the state of physics being deplorable.  It is precisely empiricism that has lead to the current situation in physics. 

 

What is ITOE?  (I'm assuming the last two words are "Objectivist Epistemology…"?)

 

The description you give about "where understanding the linguistic nature of concepts come in," seems to be precisely what Godel proved was not the case (that mathematics was a matter of the extrapolation of man-made concepts).  Logic is what you can use FROM 1+1=2.  But I am talking about how you get there in the first place.  Neither logic alone, nor experience alone, nor the two working together can get you there!

 

You say "the meaning of a concept is its referents."  What is the referent of the number 1?

 

You lost me at the bottom of your post. 

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Eiuol, I agree that we are made aware of our intuitions by experience.  But they go beyond what sense experience alone can provide.  The sense in which they are intuitive is that I can draw a simple machine on a piece of paper, show it to a child he will be able to tell just from looking at the paper and thinking about it, not only what will happen when I build that machine, but what MUST happen!  Sense experience can never tell us what must always happen.  It can just tell us what we've sensed.

I wasn't asking if we're made aware of our intuitions by experience. I was asking if experience is required for an intuition to even form. If intuition requires experience, as I'm suggesting, then by definition it isn't a priori knowledge that requires no experience at all. What I'm not claiming is that experience is sufficient for an intuition. Experience is necessary but not sufficient for most knowledge I'm suggesting that a child can only abstract away particular details after already having acquired relevant experiences. If simple mechanics are a priori, then you'd say that a child can know what a wheel would do without ever seeing things that roll - if "knowing" isn't regurgitating facts.

 

Are you maybe suggesting that all abstract thought is a priori reasoning? I once argued with someone who did say abstract concepts are about a priori reasoning that could, for example, be rooted on intuitions.

 

I'm saying that abstract thought needs to be connected to perceptual experience in order to constitute valid knowledge and concepts. Objectivist epistemology takes it that way, and the embodied cognition view takes a similar stance.

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Eiuol, you yourself in your first post here said a priori was referring to "what you can figure out from logic or pure abstraction without reference to the real world."

 

I don't believe any philosopher define a priori as "that which babies know the instant they are born."  =)

 

I would say that at least in practice experience is necessary intuition to conclude anything--even if its just the experience of thinking.  But, in the real world, clearly our conclusions arise out of a combination of sense experience and thought, both of which children progress in as they grow.  But thought about sense experience itself can't produce the kind of knowledge that we have without something else. 

 

I'm not sure I would say "abstract thought is a priori reasoning."  I think what I would say is that the essence of thought it abstraction, and abstraction has to be done according to a priori categories (concepts which cannot be given by experience).  For example, shape or extension.  To begin naming and recognizing objects out of the mess of sense perception, you have to abstract according to shape / extension.  But that concept is not given by sense experience (all that's given by sense experience is its own content). 

 

As for how we explain where it comes from, I don't know if I could venture to say.  Embodied cognition seems to be one viable theory.  It could simply be a matter of the way our brains and bodies are.  Almost certainly that has much to do with it.  But I think there is real mystery here, and philosophy is always better off being honest about what it does and does not know, and I feel that much error has come about when philosophers have substituted dogmatic ignorance for wonder (in the sense of "I don't know what this is, therefore it cannot be real.") 

 

And the conversation to me seems to have been shrouded in very loose thinking--such as the metaphor that the mind is a blank slate.  That's just a very loose, sloppy metaphor that has no place in philosophy, to my mind.  The mind is not a slate--it is a mind (which is something both complex and mysterious, unlike a slate).  You can begin to describe the difference between two very different things--a mind and a slate, by describing the a priori categories by which the mind abstracts (at least, so long as you don't believe that all knowledge comes from sense experience, which practically amounts to a nullification of all the properties that minds have which slates do not).

 

Plasmatic, I do consider myself a Popper fan.  He seems to me to be right about a lot, and one of the few people to have understood the problem of induction.  I don't agree with everything I've read though.  

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And I maintain what I said about what Godel proved meaning that mathematics was not man-made (which is to say, "merely axiomatic" or formal).  That's how he himself interpreted it and that interpretation is widely accepted in those who have not invested too much publically in the opposite ideas to consider following the argument to change their minds.

For what it's worth, I think you're right. I wouldn't use the word "man-made" here, and it'd be too off-topic to get into that right now. But it's true, math is not "merely axiomatic" or formal. In other words, math is not at root a set of provable axioms. Godel showed that no axiomatic system can prove itself. A different basis is needed than formal statements. A sensible basis might be something perceptual for the same reasons Rand thought valid knowledge is based on perception - we're not going to produce knowledge out of formal statements alone. You could logically argue that intuitions are a valid basis, too - the point of the thread.

"I don't believe any philosopher define a priori as "that which babies know the instant they are born."

And I never attempted to use that way of defining a priori. I mentioned a child because you said a child would be able to tell how a simple machine would work from only a drawing. Aside from the fact you didn't test your theory (how are you so sure they can?), it only shows that children are able to read diagrams. It also might show children are able to combine concepts to develop abstractions. Those combined concepts may have also been developed that way. We're still left with the same questions: for one's earliest concepts, is experience necessary? If so, then intuitions are not a priori - they need some amount of experience. If not, then some type of content exists before experiencing anything in a particular domain.

"Content" is the word I use there because I distinguish it from mechanisms. By mechanism, I mean some inherent cognitive ability. You can tell already think the mind is a blank slate, but I do not mean an empty mind, I only mean empty of content. Consider how, as an analogy, a digital camera takes pictures. At first, it has no photos, but it has the mechanisms to take photos. Press a button, then the camera has content. More mechanisms interpret or manipulate the photo, clearly unable to manipulate anything before taking a photo - unless it had pre-uploaded data. With my analogy, I'm saying that experiences are needed for the mechanisms to do their work, to create knowledge. If the mechanisms create knowledge, then all knowledge is connected to experience. Maybe you're claiming the mechanisms are a priori? If so, then I'm disputing that a priori categories exist, not that a priori mechanisms exist. I hope my explanation makes the discussion less sloppy.

What exactly is an a priori category then? Is it a full-fledged concept disconnected from experience? Is it only an in-born sense of organization like Gestalt principles, that are used automatically?

Keep in mind that I am not suggesting it's possible to passively observe and knowledge is impressed upon one's mind. To be more explicit about Objectivist epistemology, and I happen to agree, Rand is specific enough to say passive observation does not produce knowledge. I'll look up a page number/reference if you want.

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Eiuol, given the way that I define a priori, I don't believe the necessity of experience for forming knowledge in any way entails that that knowledge is not a priori.  I believe that it is both true that we have a priori intuitions which underlie much of our knowledge, AND that those intuitions are awoken by experience.  Again, the distinction (as I define it) has nothing to do with the conditions under which knowledge comes to arise--rather, it is about its justification. 

 

The comment I made about babies was meant to show this.  If a priori was a matter of the conditions under which knowledge arises (and was inconsistent with any knowledge that arises from experience), then newborn babies would necessarily have as much a priori knowledge as adults (since the thing that separates adults from babies is experience, and if experience caused the knowledge-difference between the two and yet was inconsistent with a priori knowledge, it would follow that babies would have to have all the a priori knowledge which adults do). 

 

Your distinction between content and mechanism seems reasonable.  Here's what I think I would say: a priori knowledge is something like a mechanism, but it can also produce content, just like looking at a photo from a digital camera tells you something about the photo's subject, but also something about the camera itself.  In the same way, through experience the mechanism which is our mind can not only tell us about the content of sense experience (which, again, consists only of itself), but other things (for instance, logic).  I think we have real knowledge (for instance) that a contradiction cannot be, and this is plainly not something given to us by sense experience--even if experience involves the conditions under which it arises.  But it can never be justified by sense experience. 

 

You asked, "what exactly is an a priori category?"  My answer is that I don't know.  There are many things that we know by association without knowing what exactly they are--including matter, people, beauty, and just about everything in the world!

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So you agree that experience is necessary to form knowledge. That means knowledge without experiences can't be knowledge since it lacks a necessary aspect. But then you say it doesn't entail that knowledge isn't a priori? You're already saying apart from experience there is no knowledge. All I can do to make that consistent is to say some knowledge isn't "formed" but arises in non-deliberate ways. Intuition seems to be your answer. To form knowledge needs experience, to have knowledge might passively appear. I don't see how passively-arising knowledge is valid or justified. Perhaps it -can- happen, but it's no good. Not reliable or worth calling knowledge. I'd still say some experience is needed even for this woozy passive knowledge.

I don't really follow the babies part. I mean, babies have experiences too. Unless you've got some knowledge of cognitive development that I don't, your simple mechanics example makes assumptions of how and if children or babies are able to read diagrams.

"I think we have real knowledge (for instance) that a contradiction cannot be, and this is plainly not something given to us by sense experience--even if experience involves the conditions under which it arises. "
Make an argument. It's not plain to me. I think it's important here to explain how facts of logic don't require some experiences to justify as true, as well as how you can know the law of noncontradiction without any experiences. It's true, you can't justify ONLY by experience, but, as you said, experience is required nonetheless.

"You asked, "what exactly is an a priori category?"  My answer is that I don't know. "
Then don't introduce an idea you can't explain! :P

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Euiol,  at times I feel like we're close to agreeing, and then we keep getting stuck on the same points!  It seems that we keep coming back to the argument that knowledge cannot be a priori if it arises through experience.  But I feel that I've repeatedly accepted that experience causes a priori knowledge to arise, but that I define it as that which cannot be justified by sense experience alone.

 

I'm not sure how to make myself more clear.  I think in every post I've made thus far I've mentioned the fact that experience causes a priori knowledge to arise, but that the key distinction (in my meaning) is not what causes it to arise, but the basis of its justification--which, in the case of a priori knowledge, cannot be sense experience.  I even quoted your initial email, in which you yourself said that a priori "isn't referring to inborn knowledge. It is referring to what you can figure out from logic or purely abstractions, without reference to the real world."  Assuming that "figuring something out" is itself a type of experience (and one which certainly occurs only in adults whose minds are well-versed in a lifetime of sense experience), this is saying something very close to what I'm saying. 

 

I'm not clear where you're getting the idea of "passively arising knowledge," or how it relates to what I'm saying.  The distinction I'm trying to make it between:

  • Knowledge that can be justified by sense experience alone (a posteriori knowledge, which includes only the content of what we experience)
  • Knowledge that can't be justified by sense experience alone (a priori knowledge)

 

Here's what I mean about babies.  If we define a priori in the way that people keep suggesting, as "that which we come to know without having any experience," then babies would have as much of it as anyone.  Their lack of experience would not matter, since this suggested-definition explicitly defines it to be independent of (and perhaps prior to) experience. 

 

But I don't define it that way, nor (I think) does anyone. 

 

To my admission that I don't know exactly how to explain the existence of a priori knowledge, you said, "Then don't introduce an idea you can't explain!"

 

But we can argue for the existence of something (and I think even have certain knowledge of the existence of that thing) without being able to say exactly what that thing is in its essence.  I think we know that we exist, but the idea of the self and its essence is wrapped in mystery and even paradox.  Matter is another familiar example.  We know it exists by constant experience, but even scientists cannot say what--at bottom--it really is, and all their explanations (to date) end in more mystery and paradox.  

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Spiral, I don't believe you understand the problem of induction.  "If you see one apple, and then another (etc.)," you say, "I hope you can draw the conclusion pretty fast!"  But if you meet one Dutch person who is tall, and then another, and then another, at what point are you justified in drawing the conclusion "Dutch people are always tall," in the same way you can draw the conclusion, "1+1=2"?   Experience + induction can NEVER lead to a universal statement (such as those of mathematics and science).  I think the ambiguity in your explanation in contained in the way you're using the phrase, "logical leap," and this will be clear if you try to define it precisely. 

 

Yes, I do and I already explained your example in the other post.  I must not be making myself clear for you to retool the same example.

 

In this case you are NEVER justified in determining Dutch people are tall as the total body of our knowledge contradicts such an absurd statement.  We know species can come in any size or form. 

 

Of course Experience + induction is not enough.  Like I have said, you have to THINK too. Make decisions.  Reference what you already know and look for contradictions. Induction requires inducting into the total field of what you know. 

 

Logical leap is just a poetic way of saying when your confident to go from what is possible to being certain you have fact

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Let me rephrase your position as I understand it, in terms I prefer. Footnotes for my opinion are at the bottom. I want to make sure I got it all right.

Although all formation of knowledge requires experience, the justification as true does not always require experience. Math is justified a priori for example. We know this because Godel proved that math is -only- a matter of extrapolating concepts*. All we can do is talk about them as universal, but not as foundational or formal proof, and experience alone won't provide valid propositions that always apply**.  Like math, science is a priori because its foundations, while they arise from observation, cannot be justified by experience alone. Further, observation won't tell us whether our statements are true or false, so we don't need observations to determine the statement's truth value.

One a priori way to justify and make discoveries are thought experiments, which of course comes through imagination. Generating the thought experiment is not done by using immediate observations. A thought experiment, by means of logic, will then justify our statements about science by demonstrating consistency. Without some method like this, claims about science go too far. So, good science and statements are justified a priori.

Simple mechanics, as a way to think about simple physics, is intuited a priori, namely, that a person justifies an answer a priori by using intuitions that aren't experiences #. Show a child a drawing, or an adult, and they'll at least get an answer with justifications.

*but really it only means you can't prove an axiomatic system with its own system, so this point actually shows you can't justify math systems by formal statements.

**You can't get universality if you mean "always true across all time", there is always a constraint of reality and within the purpose of your thinking. So arithmetic is just really abstract, but within its context, like all knowledge, it's always true.

# How does an intuition develop without being filled with all kinds of experiences? I fail to see how intuitions are not experiential, let alone as mini thought experiments.

 

"To my admission that I don't know exactly how to explain the existence of a priori knowledge, you said, "Then don't introduce an idea you can't explain!""

I said "a priori categories", not "a priori knowledge". It was too vague and undefined, and you couldn't explain or theorize what they are. If they're even real. Why are you justified in saying they're real?

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