We do not intuit time or space; we intuit objects in time and space. To intuit is to follow from sensation, and the sensation proper of time or space is impossible, insofar as both are conceived as infinite and empty. — Mww

I'd say "to intuit" is not well defined, and people use "intuition" in various different ways which are pretty much all quite vague. Intuition may refer to immediate apprehension by the mind, or it may refer to immediate sense apprehension, perception. The latter would follow from sensation, but the former not necessarily so.

There's a further problem with the concept of "conception". If conception is understood as something the mind does, to create a concept in one's mind, or if conception is understood as receiving a concept from somewhere else, these two different modes of understanding produce different ways of apprehending "intuition".

Our discussion concerns the apprehension of space and time, and I've already cautioned you about the need to separate these two, i.e. not to place them in the same category as both being apprehended in the same way. We can see that "space" is fundamentally conceived out of necessity, it is necessary that we allow that there is something called "space", the medium within which objects exist, allowing them to move around. The conception of "space" is necessary to allow for the reality of independent objects.

But the issue of how we conceive of "time" is far more difficult. This I believe, is because we already have a sense of before and after, within us, by the time we are born. As we grow it manifests as the difference between memory of the past, and anticipation of the future. But I believe a fundamental respect for a difference between future and past is already inherent within the constitution of living beings. This supports a more specific sense of "intuition" which is based in what we know as "instinct".

So I think we apprehend the difference between future and past in a way which is not the product of a logical necessity like our apprehension of space; the concept of space being something required to make sensation intelligible. It is already deep within our constitution as a base for fundamental disposition, and the problem is that we cannot call this a form of understanding. I can say that I intuit a difference between past and future, but I cannot say that this is derived from sensation, nor can I say that it's a product of any logical reasoning. It's something I've received, or simply have, but I do not understand it, nor can I conceptualize it, or sense it in any real way.

Second, the most basic mathematical principles are subsumed under the schema of quantity, not relation, or, in your terms, order. — Mww

I thought the same, but I was corrected in this forum, where there are many participants who are well versed in math. If you look at mathematical axioms you'll see that cardinals are derived from ordinals. And even as far back as Plato there was questioning concerning the relation between quantity and order. Ultimately, we can see that One was understood as first, rather than as a quantity.

When we are very young we learn to count, and counting is first learned as ordering, two comes after one. This is what makes "numbers" so consistent with platonic realism, the number itself is seen as an object with a place in an order. If a number simply signified a quantity, we'd say the numeral signified the quantity, and there would be no need for "the number" as an intermediary between the numeral and the quantity. But since the numerals are used to signify an order, then we have to assume something which occupies that place in the order, hence "the number". Then one comes before two, and two comes before three, etc.. These are not the symbols we are talking about, but the assumed numbers themselves, which have that order, and the symbols, the numerals, represent the numbers.

True enough. But if it is the case we don’t function at all, in any way, shape or form, when we dismiss the basic principles of logic, then it is reasonable to suppose we couldn’t do that in the first place, insofar as we must use them in order to assume their dismissal. It follows that if we cannot assume to dismiss them, we are left with merely getting them as correct as we can. — Mww

I don't agree with this at all, and I've argued it in many places in this forum. We do not need to assume replacement principles to reject principles which we find unacceptable. This is fundamental to skepticism, we can reject principles without replacing them. And, rejection itself need not be principled. So for example, if I refer to this fundamental instinct, this "feeling" which I have concerning the difference between past and future, I cannot understand this instinct, nor can I formulate it in a principled way. However, I can reject propositions based solely on this "feeling" which I have. So we can reject propositions based on "feelings", and dismiss them, though the rejection is fundamentally unjustified, therefore not properly principled, and others might call this rejection "irrational".

Sorry, but the nature of logic is in judgement, not intuition. — Mww

That's not what I said though, I said it was grounded in intuition. And the grounding of logic, substantiation, what makes validity work for us, is fundamentally different from logic itself. The grounding has to do with the relationship between logic and our world. "Substance" is what grounds logic, but it is not a part, or characteristic of logic itself.

Think of it this way: you know how when we perceive something, when we are affected by some imprint on the senses, we are never conscious of the information that flows along the nerves? We sense the beginning, we cognize what the beginning was, at the end, in the brain, but all that between, we know nothing about whatsoever. THAT is intuition, in the proper, albeit metaphysical, sense. And because we are never conscious of our intuitions, but we are certainly conscious of the judgements we make on our sensations from which the intuition is given, and logical determinations are the objects of judgements alone, it follows necessarily that intuition cannot stand in any relation to the nature of logic. It may be said intuition is the ground for the possibility of logical determinations, but that is not to say they determine the nature of logic. — Mww

So I think that what you say here about "intuition" is consistent with what I said about intuition grounding logic, if you understand the difference between logic itself, and the grounding of logic. But I obviously do not see the basis for your conclusion "it follows necessarily that intuition cannot stand in any relation to the nature of logic". Remember that standing in relation to something is different from being a part of that thing.

What is at issue is your final phrase here, "but that is not to say they determine the nature of logic", and how we understand "the nature of logic". What is the nature of logic? Where does it come from, and how does it exist? However one answers these questions, and there are different possible answers depending on one's metaphysics, dictates whether this proposition is judged as true or false. If logic is something produced by the conscious mind, for the purpose of understanding all this information you describe as being between the thing perceived, and the perception of it, what you call "intuition", then the exact opposite of what you state is the case. Logic is something created by the conscious mind, shaped and formed for the purpose of understanding intuition, therefore intuition determines the nature of logic.

It is far too common, for people to attempt to associate logic to an external world which is the object of perception. Then they will try to see how logic works to understand this external world. But that is to ignore all which is intermediate, what is between logic and the world, what you call "intuition" here. The problem of course being that we cannot apply logic directly to the external world, all we have is the in between, the information received, the intuitions, to apply logic to. So even if logic is something created by human beings for the purpose of understanding the external world, we are stymied in our attempts to apply it because all we have is intuitions about the external world to apply it to.

Besides....there are those occasions when we employ logical principles even without an intuition, without an object making an impression on the senses. Case in point....the guy that invented the Slinky. Sure, springs and stuff falling are sensuous impressions, but you can’t get a Slinky as such, from those two intuitions. To connect those into an object that doesn’t yet exist requires more than the antecedent intuition of each. In just the same way, you cannot get to 12 if all you have is a 7 and a 5. — Mww

This is why we need to distinguish internal intuitions from external intuitions. This I think is very important. If we simply say that logic gets applied to intuitions, and if internal intuitions are fundamentally different from external intuitions, then we'd need different logic for internal than we need for external.

Kant, I believe outlined this division, space as the condition for understanding external intuitions, and time as the condition for understanding internal intuitions. The senses are the principal medium in the external intuitions, and emotions, or "feelings" (in that sense) are the principal medium for internal intuitions. The internal intuitions are not well understood, because the observations required for science are difficult. Internal intuitions are often compared to, and even described by the same terms as the external, internal "sensations", and internal "feelings" for example. But we do have a clear division in definition of "intuition", between immediately apprehended by the mind, and immediately apprehended through the senses.

Maybe, the red pill and the blue pill actually consist of the very same thing?

Actually though, I think there is a very real difference, which involves the relationship between space and time. As explained in my post, intuitively, time is logically prior to space. And for any person who takes a few moments to contemplate this question, in meditation or any other mystical practise, this intuition starts to become very clear. But in the science of physics, this order has been reversed, such that space is prior to time. Therefore we can conclude that our true intuitions must be suppressed for the sake of adopting the time/space relationship employed by physics.

When presented with this problem, we can proceed toward resolution by assuming that our basic intuition of time is wrong, or by assuming that the representation of time employed within physics is wrong, (or both). But if we assume that our fundamental (base) intuitions are wrong, then we have nothing left to go on. We must dispose of the most basic principles of logic, such as identity, and non-contradiction, and we are left with zero, nothing as a starting point. Any chosen starting point, for any sort of understanding, would be completely arbitrary and randomly chosen.

This is what "pure mathematics" gives us. It appears like we can choose any arbitrary starting point (axiom), and produce a synthetic conceptual structure based solely on One point. Order is not necessary, we can start at any given point. However, the nature of logic, and it's ground in intuition, really demonstrates to us that this is untrue. In reality we need more than one starting point. And, we need an assumed relationship between the plurality of starting points. If the relationship between the multitude of points is not assumed to be one of necessity, the entire structure has no logical stability.

In turn, this reality show us something about "intuition" itself. It can be neither a priori nor a posteriori, but it must necessarily be a combination of both. That is what produces reliability.

But to 'explain reason' is to invariably sell it short! As soon as you account for it in anything other than it's own terms, then you're denying the sovereignty of reason. I'm beginning to suspect that the very existence of reason is actually an inconvenient truth for a lot of analytical philosophy. — Wayfarer

This is why Banno characterizes (defines) "reason" as "a group enterprise", rather than as the activity of an individual mind. Instead of describing reasoning as something which an individual mind does, Banno describes reasoning as having the essential property of "language" (requiring, or needing language). This makes language logically prior to reasoning.

Now, instead of the true description, in which individual reasoning minds use language as a tool, Banno has a group of people involved in an activity called reasoning, and the group are using language as their tool. The obvious problem with Banno's argument is that it is utterly impossible to locate, identify, and understand this group activity, called reasoning. Every time that we try to find and identify the activity of reasoning, we see very clearly that it is the activity of an individual reasoning mind. Therefore Banno's argument is based on a false description; the unsound premise that reasoning is a group enterprise.

It's usually called collective intentionality — Banno

Take the following passage and replace "intersubjective" with "collective intentionality", and tell me how is "collective intentionality" supposed to resolve the problem you said "intersubjective" has.

"Intersubjective" is one of those oxymoronic terms that folk use to avoid thinking. If the subjective is private, then the notion of the "inter" subjective makes no sense. If the subjective is not private, then adding "inter" to it is superfluous.

Nothing is solved by waving such a word around. — Banno

All you have done is replaced one oxymoronic term with another. Nothing is solved by waving "collective intentionality" around.

How boring. The real fun starts in going the other way. Everybody thinks; no one knows how thinking happens. So...there ya go, ripe for theoretical musings. — Mww

Well, I'll qualify my statement then. Instead of saying that there is no such thing as pure a priori, I'll say that if we seek such, we find a big division between the intuition of space, and the intuition of time. And this division manifests itself in the principles of mathematics. Time is fundamentally order, and space is represented by lines. But order is of discrete units, while a line is a continuity.

Space, being an intuition dealing with how we relate to the external, cannot be a priori. This is because it can only be created from the individual seeing oneself as an individual, distinct or independent from one's environment, and this condition is posterior to the most primitive experience of the human being, being born. If we look back to the prior condition, the earlier condition, which is being in the womb, at that time the developing individual is united to, as a part of something larger. So there is no self with an external, as the thing which will be a self, is simply an internal part of something else. This is the condition which some mystics guide us toward, the position in which we are simply united within a grander whole, and there is no proper individual self, and no proper "external".

From this position, there can be no intuition of space, as this the original condition, in which there is no external, therefore no space, and this experience is prior to the intuition of space. And this is why, within the manifestation of mathematics. spatial conceptions lack in necessity. Geometrical principles can be altered as we see fit, so that, for example, parallel lines may meet in a curved spacetime. Our spatial intuitions, which lead us into geometry are contingent on how we conceive of time, which is derived from our temporal intuitions.

This places the intuition of time as deeper than, and prior to, the intuition of space. It manifests as the most basic of mathematical principles, order. The issue which Wayfarer points us to with this thread is the question of necessity in order, which is understood through "causation". But the real question is whether necessity can be removed from order, in a way similar to how I described that necessity can be removed from spatial conceptions.

Modern mathematics uses axioms which deny the necessity of order (a set of objects without an order for example). To validate such an axiom, "order" would have to be contingent on an even deeper intuition. We'd have to intuitively apprehend something deeper which order is dependent on. But I cannot find any deeper intuitions, to say that order is dependent on something else, and validate removing necessity from order. So I find that such axioms which attempt to remove necessity from order, are extremely counter-intuitive. How this intuition of time, manifesting as order, is itself grounded, whether it is grounded in experience, or something more fundamental than experience, as prior to experience, and a condition for the possibility of experience, is probably an issue of how we define the terms.

I remember. I generally agree, taking exception only to your referring to “pure physics” in a Kantian context. As brought to light by ↪waarala, it is clear there is a pure part of physics with respect to the a priori principles which make the science possible, but “pure physics” as a general conception, has not the same distinction as....

“...Before all, be it observed, that proper mathematical propositions are always judgements à priori, and not empirical, because they carry along with them the conception of necessity, which cannot be given by experience. If this be demurred to, it matters not; I will then limit my assertion to pure mathematics, the very conception of which implies that it consists of knowledge altogether non-empirical and à priori....”

....in which we see how he wishes “pure” regarding the “theoretical sciences of reason” to be understood. — Mww

I believe there is no such thing as "pure" a priori. The a priori is always conditioned by the basic intuitions, space and time, which are inherently dependent on experience. Even in mathematical principles, if we attempt "the pure", we remove ourselves from an applicability with the consequence of useless fiction.

The US could not have stopped the invasion. Just because you know something is brewing, doesn't mean you can stop it. — Olivier5

I think the US surely could have prevented the invasion, by submitting to Russia's demands, and pressuring Ukraine, and all NATO countries to submit to Russia's demands. But we do not know the full scope of Russia's demands. On the surface it appeared as the demand that Ukraine stays out of NATO, but I'm sure it wasn't so simple, and there was much more below the surface. So the issue of whether it would have been wise for the US to collaborate with Russia, and prevent the invasion, is another question altogether, regardless of any far-right extremism which Isaac mentions (which is really irrelevant as an internal matter).

Notice that these synthetical a priori principles are called "judgements". This is why skepticism as an approach to fundamental principles is very important, as a tool to find mistakes within these judgements. Without skepticism, these synthetical judgements are simply taken for granted as "discovered" principles like Hilary says, when in reality they are synthetical structures created by judgement. Plato is often misunderstood as promoting a position in which we take these principles for granted as eternal truths, when in reality Plato was a skeptic demonstrating the need to question such principles.

Galileo counters the Aristotelian approach not by performing experiments, but by showing that it [e.g. the mathematical fabric of space-time] must be so and not otherwise. In this sense, physics is made to be an a priori discipline of necessary truths. Koyré sums it up as follows: ‘The Galilean revolution can be boiled down … to the discovery of the fact that mathematics is the grammar of science. It is this discovery of the rational structure of Nature which gave the a priori foundations to the modern experimental science and made its constitution possible.

That, I think, is the source of Kant's conviction that physics can be an a priori science - that 'physics, like mathematics, is a body of necessary and universal truth.' Noble sentiment but hardly sustainable in respect of physics since Einstein, I would think. — Wayfarer

When mathematics is "the grammar of science", which formats the way that the structure of nature is revealed to us, then what is understood as "the structure of nature", the phenomenon which bears that name, is literally formed or created by that mathematics.

The issue then becomes the quest to maintain correspondence between "the structure of nature" (as created by the human mind), and intelligibility (as dictated by what the mind can understand). When mathematicians allow the axioms of "pure mathematics" to stray outside the limits of the fundamental laws of logic, identity for example, then there is inconsistency between the grammar of mathematics and the grammar of intelligibility. Since "the structure of nature" is based in mathematics, it is possible that this structure may become completely unintelligible to us, depending on the intelligibility of the axioms employed.

That's quite a truthful graph. Calling it an "Central Bank bubble" is quite apt. — ssu

Why do they position the "bubbles" after the bust, instead of putting them in the proper place, before the bust? I wouldn't call that truthful.

My instincts, however, inform me that he's not entirely correct about the relationship between language and philosophy. — Agent Smith

I think he's actually quite wrong, having a number of things backward. He's too enveloped by the idealist tradition.

.such as property, money, government, credit...? — Banno

These things are not created by language, they are created by human beings, with language as a tool. Remember the principle, "meaning is use". This makes language something used, like a tool, as the means toward various ends. When we use tools to bring about the existence of things, it is incorrect to say that the tool is the cause of existence of the artificial thing, the tool is just the means to the end.

This is the problem with platonic realism which Wittgenstein helped to expose. If we tie the symbol directly to the idea, as if we were naming an object, like when we say the symbol "2" represents a mathematical object, the number two, then the idea must be something eternal and unchanging, making it passive without causal capacity. This issue is commonly presented to dualists as "the problem of interaction".

We can avoid this commonly cited problem of dualism by portraying symbols as tools being used for various purposes, rather than as necessarily the name of an object (Idea). But then we have to account for the reality of intention, apprehending intention as having causal influence in the world. This is what is known as final cause.

That would be a total impossibility. For the simple reason that the second universe would be part and parcel of the real universe. — Ken Edwards

Then for what reason do you call it a second universe. If it's just a part of the real universe, without anything separating it from the real universe, then the designation of "second universe" is not justified. It's a part of the one.


I can more easily agree with the position you present. The issue I see is that there is a multitude of "universes", each being the presence of one person's own inner world. And since "multitude" implies a number of distinct individuals, there must be something which separates us all. This would be the external world. (I'll use "world" so that it's not confused with the "universes" which are internal). But this external world which separates us must be just as real as the internal universes, or else there'd be no separation between us, and no multitude of universes.

I call this external world the medium. I think it's better to use the singular "medium" rather than the plural "media", because it is essentially one, and this one thing separates all the individual minds which are many. The medium though, as matter, seems to consist of numerous different forms, and this is what gives the illusion that it is a multitude.

I've walked away from a couple of jobs because the bullshit was just not worth my time. — L'éléphant

And after a while you start to ignore the bullshit and just do the job.

Mostly in my dealings with people. — L'éléphant

I find the very opposite. My dealings with other people cause me a lot of stress, and keep me awake at night. Then I need to deal with myself, so I pick up some good philosophy to read and I fall asleep almost immediately. So philosophy is really good for dealing with myself, by allowing me to ignore my dealings with others.

I find philosophy has been very useful toward getting a good night's sleep.

the solution is to develop advanced defensive weapons that can automatically spot, track, and disarm/shoot down any nuclear weapon missiles. — Christoffer

What would happen to a nuclear missile that got shot down?

All I really want to say here about the Second Universes is the following:

"My second universe is smarter than your second universe".

Ken Edwards — Ken Edwards

OK Ken, I take that as a challenge. How would we decide the truth of this, by our capacity to effect change in the real universe? I would say that the smartest second universe is the one that can exist without having any effect on the real universe. Want to try?

In my scientific knowledge there is no gap — Hillary

What the hell are you doing on a philosophy forum, if you already know everything?

I advocate that they surrender to a dictator who wants to secure his regime against foreign interference (and is willing to use brutal force to do so). Again, your personal assessment of the situation is not a fact, its an opinion, one with which I, and many experts in the field, disagree. — Isaac

In this world of globalized trade and international relations, within which we live, what kind of isolation would be required to secure one's regime against "foreign interference".

"Foreign interference" is just an ill-defined catch phrase, which in most usage on this thread, only means someone else is getting in the way of me getting what I want. The act of getting in my way, is carried out by some foreign individuals or agencies, and by reference to some laws is deemed as illegal. So for example, Hillary Clinton complained of foreign interference in the American election.

As you can probably see, "foreign interference" as an illegal activity (if we could separate it from the legal sort) is a shifty sort of phantom, like a mirage. It has a very real existence, but its physical presence, i.e. where it originates, or its cause, is not what it appears to be. How do you think that the use of "brutal force" could be effective for defending oneself against such an entity as "foreign interference", whose mode of existence is ideological rather than physical?

It appears that a time will come when even a zygote will be viable ex-utero. What then? Abortion would immediately have to be made illegal, oui? — Agent Smith

I don't think so, pregnancy will then be obsolete. The only babies being produced will then be designer (GM) babies. Human abortion will be enforced by the newly derived species, resulting in the extinction of the human species.

Lula said Biden and European Union leaders failed to do enough to negotiate with Russia in the run-up to its invasion of Ukraine in February. — https://www.reuters.com/world/americas/brazils-lula-says-zelenskiy-as-responsible-putin-ukraine-war-2022-05-04/

What does this mean? One ought to give concessions when force is threatened.

Let me try again.

It's quite simple. Insofar as men are 100% responsible for all pregnancies, if a woman falls pregnant and carries to term, the man takes 100% responsibility for the child.

The abortion debate would stop tomorrow. — Streetlight

By not being able to impregnate women, well, unless they plan to, with them.
Problem solved, no more abortions. :up: — jorndoe

Neither of these ideas, I believe, could resolve the problem. It appears like the problem which leads to a never ending abortion debate, is that many people do not understand the true reality of "failure".

If we represent the birth of a child as the successful end product of the sexual act, then a true representation would show failure as far more prominent than success. Further, if we proceed toward representing this as a designed process then we need to allow that failure is essential to the design. The prominence of failure is what renders the individual successes as something special.

Religious zealots who refuse to recognize the reality of failure as an essential part of the design, have no capacity to understand evolution and the very important information which the reality of failure gives us concerning the nature of the design.

When good is measured only by success, with complete disregard and disrespect for those who have failed, then we have a world full of uncaring, uncompassionate people who view mistake as unforgivable.

It's quite simple. Insofar as men are 100% responsible for all pregnancies, if a woman falls pregnant and carries to term, the man takes 100% responsibility for the child.

The abortion debate would stop tomorrow. — Streetlight

By not being able to impregnate women, well, unless they plan to, with them.
Problem solved, no more abortions. — jorndoe

Sorry that was a premature ejection of the post.

why not have males carry some responsibility here, instead of males just legislating females' bodies? — jorndoe

What do you think, we might implant the males with a womb, and make each one of them take a turn at looking after the unborn? Or how else do you propose that the male might carry some of this responsibility?

Males that don't express they want children, yet are interested in sex, get a reversible vasectomy (or something to that effect). — jorndoe

If you think this way, why not go all the way? Once they express interest in sex, it's already too late. Better to make it a practise as soon after birth as possible, just like circumcision. And forget about reversibility, slip an elastic band on as soon after birth as possible, like they do with bull calves. Masculinity is the scourge of humanity (maybe this should be in the Putin thread). Go back to Plato's Republic, only the genetically superior babies (created in a lab in the modern day) will be allowed to see the light of day. (Nazism?)

Freedom in arithmetic? Contradiction.

Some bees are just lazy. — Cuthbert

Those are the drones. They are born to be lazy. But they have perhaps the most important job, to fertilize the queen. I guess they are given the privilege of being lazy, to ensure that they willingly carry out the one very important job that they do have.

It's just a tricky way of saying, that there's a way that the distance between two objects increases at a rate which is faster than the speed of light. It's allowed to be faster than the speed of light, because the change in distance is not called "motion". The concepts of space and time, which are the basis for the concept of "motion" are tied to relativity theory in a way which makes understanding this change in distance as a "motion" impossible. So they are forced to say that this change in distance between two objects is something other than "motion".

Philosophers have valiantly fought the tyranny of religion and tradition; but they have come under the sway of the tyranny of knowledge and become hypnotised by the tricks of science. We are ruled by arrogant certainty and it is taking us to our grave. Don't be so sure. — unenlightened

Those are the words of God Himself! I'm sure of it. Look, He can alter the truth of fundamental arithmetic, He made the poll scores add up to 101%. He\s more powerful than Google!

Other: Unenlightened believes Unenlightened is God.
BTW, so do I

The uniqueness of an individual's behaviour goes far beyond the present capacity of the human intellect. Consider that the human being does not even have the capacity to understand the unique behaviour of a basic inanimate particle like a photon, or an electron. Since the activities of these basic particles have a great affect on the activities of larger things, it is very obvious that human beings do not have the capacity to understand the uniqueness of an individual being's behaviour.

Doesn't that make premise 2 and the conclusion obsolete? — Magnus

No, I think it's the inductive principle which justifies premise 2 as true. Premise 1 says that if a thing has a beginning, it has a cause. Premise 2 says that "the universe" is that type of thing.

Your discussion concerned "the universe" being defined as a collection of all things. But that is misguided (or a 'strawman') discussion, because that is not how "the universe" is defined here. "The universe" is defined as a thing which has a beginning.

Whether or not premise 2 provides an accurate description of "the universe", is a question you might ask. You might also ask whether or not Premise 1 provides an accurate description of things which have a beginning. Showing that the premises are not necessarily true, demonstrates that the argument is unsound. But to give "the universe" a different meaning, other from the one used in the argument, doesn't prove the logic invalid, it just makes a strawman refutation.

If X is true for everything within the universe, then X is also true for the universe itself. — Magnus

I think the idea is that there is an inductive conclusion which is the first premise: "X is true for every thing". Then , "the universe is a thing". Therefore X is true of the universe.

The fact, if it is one, that we can only understand events by thinking causally does not entail that the events must be causal. Also, I haven't said that events can happen without cause. I have said there is no logical contradiction involved in thinking that they could happen without cause. — Janus

So let me go through the problem again. I'll try to be concise and to the point, so maybe you'll understand this time. Suppose I believe both, that events can happen without cause, and, that I can only understand events as being caused. Aren't these contradictory beliefs? Doesn't the proposition "events can happen without cause" present itself as a sort of understanding of "events", which contradicts "we can only understand events by thinking causally"?

This is why I asked, what does "understanding" consist of to you. If "understanding" consists of applying logic, and ensuring that the thing "understood", adheres to logical principles, then the proposition "we can only understand events by thinking causally", contradicts the other, "I understand that events could happen without cause". So the two cannot be a part of one understanding of "events".

This is because "event" is a word we use to refer to things, and you have defined "event" by saying that it is a type of thing which which can only be understood causally. So "event" necessarily refers to a causal type of thing. Then you turn around and say that there is "no logical contradiction involved in thinking that events could happen without cause". But clearly there is contradiction here, because you have defined "event" as the type of thing which is understood causally. This other type of thing, which happens without a cause, cannot be classed as an "event", because it does not fit in that definition of "event" as things understood causally.

You need to either change your definition of "event" to allow that we can understand "events" through means other than causation, or adhere to your definition, and place these things which can happen without cause in a category other than "events", to allow for the truth of this understanding of those things.

I don't see how critical thinking is relevant here. What paves the road to mathematical competence is the removal of critical thoughts. This is submission to authority. What is important, is that the student accepts the authority of the teacher. Then the student accepts without being critical.

What I described is a separation between the principles themselves, axioms which are taught, and this art of teaching, the technique by which the teacher impresses one's authority on the student. If the principles to be taught, the axioms themselves, are designed so as to fulfill the task of impressing authority, then they are not principles of logic, but principles of persuasion. That is, if the students are taught to see the principles, or axioms, themselves as the authority, rather than seeing the teacher as the authority, this opens the door to abuse of the students, because the principles and axioms are really the tools of those who use them. Then instead of the student learning to be the master of the tool, the student learns to submit to the tool, becoming a slave to the tool, without knowing who the master of that tool is. So the question is, are you inclining your students to be masters of the art, or are you teaching them to submit to the principles of mathematics? And various students from different backgrounds might perceive the same teaching methods in very different ways, due to the teacher's capacity to establish one's own authority, or relying on appealing to the authority of the principles.

For something to be psychologically necessary is not always for something to be logically necessary.Thinking in terms of causation may be necessary for our rational understanding of things; our rationalizations so to speak, but this is not the same as to say that thinking in terms of causation is logically necessary. — Janus

I don't understand what you are saying here. I asked, how can you say both, that causation is not logically necessary, i.e. that events can happen without cause, and also that the only way we can understand events is through causation. If some events can happen without causation, then how can causation be the way to understand these events? Do you see what I mean? If the only way that events are understandable is through causation, then the idea of an event without cause is not understandable, because if it was, that would contradict "causation is the only way we can understand events".

You need to demonstrate that a phenomenon is non natural... — Nickolasgaspar

I already did that.

I taught at the college level for many years and never thought of the subject or my teaching strategies as racist, but I know only a little of how math is taught K-9. — jgill

Your reference to "teaching strategies" indicates that you recognize that math itself is not the basic problem, but the way that it is taught may be a problem. That's what StreetlightX indicates as well. The issue is the clearest at the most basic, elementary levels. There appears to be a need to employ examples at the elementary level, and the examples are chosen, or created, with various intentions of being relevant, interesting, insightful, inciteful, or whatever. You can see that relative to different people, of different backgrounds, different examples will have different effects. If the desired effect is to encourage the student to participate and continue in the learning process, there will be discrepancies in successfulness, depending on the significance of the examples.

If we proceed toward higher levels, the issue (problem, when you see it that way) goes much deeper, and it gets much more complex and difficult to identify. The often cited issue of marking, and designating correct and incorrect is just the tip of the iceberg. I'm sure you understand that in the education of a subject like mathematics there is a very real need for direction in the form of the judgement of correct and incorrect. But we can ask what is this judgement based in, where is it grounded, and we see that "correct" means consistent with the currently accepted axioms.

Now, you'll know from my discussions in this forum, that mathematical axioms are not based in empirical truth, and I do not even believe that they are logically consistent. The mathematical axioms which are accepted into the community of those who apply mathematics, are the ones which are useful within their fields of operation, so the judgements of "correct" and "incorrect", in the teaching of mathematics are very pragmatically based. This is why the different courses of mathematical education in high school are now sometimes geared toward specific career goals, what is required for that specific field of education. It used to be that math was divided by categories like "basic" and "advanced", etc., but this was sort of degrading to the person in "basic" math, so I think the trend today is to offer the math which is designated as what is required for a specific discipline..

Consider if you will, the relationship between what mathematics is, and the way mathematics is taught. We can say that mathematics consists of tools, and the tools are accepted into use, therefore designed to an extent, for the various uses. Teaching, on the other hand is a way of manipulating minds to accept specific things as correct and incorrect. The two seem very distinct, but imagine if there is a reciprocating relationship between the two. Then the tool might be designed toward manipulating minds toward accepting specific things as correct. Historically such a tool would be known as "rhetoric". But when it infiltrates into logic and mathematics it's better known as "sophistry".

I believe that this is where the issue becomes a problem. The issue is the reality of what mathematics is. The problem is that since it's a tool to be used, it can also be used abusively. In relation to the op, the problems of the teaching strategies become a problem of the mathematics itself, when the teaching strategies become implanted into the mathematical principles themselves. That would be when the principles, or axioms are designed to manipulate minds in a specific way.

I asked you way back, weeks ago, if you minded me using "non-natural" instead of "supernatural". And you never objected to that.

Ok, rather than call what is outside of natural "supernatural", would you prefer "non-natural"? I don't really care about the terminology. If you dislike the term "supernatural" let's just call it "non-natural". — Metaphysician Undercover

Now you've just confirmed what I thought at the outset, and what I stated way back 17 days ago, when I first engaged you. You very clearly have a preconceived notion of "supernatural", and a bias and prejudice towards this notion. This prejudice disables your capacity to approach the subject logically, with an open mind.

This is clearly a biased statement.. Whenever evidence and logic indicate the reality of that which is beyond the natural, then the appropriate conclusion is the supernatural. To deny the reality of what the evidence and logic lead you toward, because it's contrary to what you already believe, is simple prejudice. — Metaphysician Undercover

I think you're misunderstanding. I'm not saying that understanding events in terms of (some kind of) causation is somehow "invalid"; in fact it is the only way we can understand events. Any explanation of the connections between events must posit some hidden forces or powers; whether those are gods, animating spirits or mechanical causes — Janus

Then how do you justify your other statement, that causation is not logically necessary?

I agree that causality is, like freedom or truth, irreducible, insofar as it cannot be explained in terms of anything else. But It is not logically necessary. There is no logical contradiction involved in thinking that events might simply happen without cause or reason. — Janus

If causation is "the only way we can understand events", how can causation be "not logically necessary"? Is there a way that we understand things (events in this case), in which we do not employ logical necessity? And how does "causation" fit into this mode of understanding?

At first glance, it would appear like if there was any sort of "understanding" which could not be demonstrated and justified through logical necessity, it ought to be dismissed as misunderstanding. What type of "understanding" is valid "understanding", yet it is not given by the means of logical necessity? We'd have to say it's not a "valid" type of understanding, yet it is still a type of understanding. How can we classify this as "understanding" when it's just as easily classified as "misunderstanding"? Or do you base "understanding" in something completely different from logical necessity? What constitutes "understanding" to you?

Well if you understand logic you would understand WHY we don't use "negations" to define things — Nickolasgaspar

You don't seem to understand that when the term being defined is the negation of another, i.e. "non...", then we refer to the other to derive our definition of the negation. For example, to define "non-drinker", we refer to what "drinker" means. To define "non-contagious", we refer to what "contagious" means, to define what "non-partisan" means we refer to what "partisan" means, and to define what 'non-natural" means we would refer to what "natural" means.

Supposed the limited capabilities as an observer do not allow you to classify something. — Nickolasgaspar

I explained that this is not the case in this instance. We know, through logic that the processes must exist. We also know from the definition of "natural" that these processes cannot ever be placed in the category of "natural" without changing that definition.

Suppose we we found out a new natural mechanism and we need to adjust the definition to include it. — Nickolasgaspar

This is contradiction. If the new mechanism is excluded from the existing definition of "natural", it cannot be called a "natural" mechanism. So finding a process which cannot be classed as "natural" by the definition, does not justify changing the definition just because you want to call it a "natural process".

Why don't you just define "natural" right now, to say that every actual thing, and every possible thing, or process is natural? I've given you that option. But I am doubtful as to how accurate it is to call a possible thing "natural".

You can not claim that there are non natural things necessary to explain our universe, while you are unable to define and show which "exotic" properties those things have that make them necessary. — Nickolasgaspar

I don't see how "exotic" has anything to do with this. If it is true that there are things in the universe, processes or whatever, which cannot be called "natural" then we can call then "non-natural". Whether or not they are "exotic" or whatever, is irrelevant.