It does not follow from the fact all sciences of reason contain synthetic a priori judgements as principles, that instances of particular relations of particular conceptions, are all principles in themselves................................If you wish to stipulate that Kant’s synthetic a priori is the principle that….that’s fine, but I doubt it’s what Kant intended for it. — Mww

Kant writes that the term "principle" is ambiguous.

B356 The term "a principle" is ambiguous, and commonly signifies only a cognition that can be used as a principle even if in itself and as to its own origin it is not a principle.

Kant writes that on the one hand there are "principles absolutely" as used in the a priori concepts of the Categories, and on the other hand there are "principles comparatively" as used in the theoretical sciences.

B358 Thus the understanding cannot yield synthetic cognitions from concepts at all, and it is properly these that I call principles absolutely; nevertheless, all universal propositions in general can be called principles comparatively.

Paul Guyer and Allen Wood in the Introduction to the CPR talk about "synthetic a priori principles", presumably as "principles comparatively".

page 13 - At this point in the Critique Kant has completed the largest part of his constructive project, showing how synthetic a priori principles of theoretical cognition are the necessary conditions of the application of the categories to sensible data structured by the pure forms of intuition.

page 85 - Synthetic a priori judgments are contained as principles' in all theoretical sciences of reason.

I agree that the Categories are "principles absolutely"

I could change the wording of my belief to "Kant's synthetic a priori is the universal principle that we can discover a priori necessity from a posteriori contingency".

Synthetic a priori is not itself a principle; it is the condition of principles, unities, conceptions and anything else to which it applies, in which representations relate to each other in a certain manner, re: synthetically, and, representations are of a certain origin, re: a priori................If you want to say certain forms of representations adhere to the synthetic a priori principle, you haven’t in the least said anything about those forms, other than give them a name, without anything about what it means to be so. — Mww

The relation between thought (a priori) and being (synthetic)

Paul Guyer and Allen Wood in the Introduction to CPR explicitly say that synthetic a priori principles can be established, meaning that there can exist synthetic a priori principles.

The "Transcendental Analytic" has prepared the way for this critique of traditional metaphysics and its foundations by its argument that synthetic a priori principles can be established only within the limited domain of sensible experience.

Yes, "synthetic a priori" is the name of a principle, not a description, in the same way that "the conservation of energy" and "the conservation of momentum" are names of principles.

Kant's "synthetic a priori" is the name of a principle that relates the synthetic to the a priori.

My original point is that B276 can be read as a transcendental argument, not in the sense of transcendental used by Kant in CPR A2, but in the sense as used by the Britannica.

CPR A2: "I call all cognition transcendental that is occupied not so much with objects but rather with our a priori concepts of objects in general. A system of such concepts would be called transcendental philosophy."
Britannica transcendental argument: "a form of argument that is supposed to proceed from a fact to the necessary conditions of its possibility."

In B276, The Refutation of Idealism, are the two statements which come under the Britannica definition of transcendental rather than the A2 definition.
1) The mere, but empirically determined, consciousness of my own existence proves the existence of objects in space outside me.
2) Consequently, the determination of my existence in time is possible only by means of the existence of actual things that I perceive outside myself.

This is why I originally proposed that “Kant's synthetic a priori is the principle that we can discover a priori necessity from a posteriori contingency".

Since then we have been discussing my expression "Kant's synthetic a priori is the principle that....."

You have been making the case that "synthetic a priori isn’t a principle", whilst I have been making the case that ""Kant's synthetic a priori is the principle that....."

This needs to be resolved before continuing with the remainder of my statement that "we can discover a priori necessity from a posteriori contingency".

.synthetic a priori isn’t a principle, it’s a relation of the content of certain kinds of conceptions to each other — Mww

You said synthetic a priori is a principle; Kant says synthetic a priori judgements are principles. — Mww

Kant writes about synthetic a priori unity (B264), synthetic a priori concepts (A220), synthetic a priori about appearances (B217), synthetic a priori cognitions (B19) and synthetic a priori judgements (B19).

Paul Guyer and Allen Wood in the Introduction talk about synthetic a priori principles.

The "Transcendental Analytic" has prepared the way for this critique of traditional metaphysics and its foundations by its argument that synthetic a priori principles can be established only within the limited domain of sensible experience.

At this point in the Critique Kant has completed the largest part of his constructive project, showing how synthetic a priori principles of theoretical cognition are the necessary conditions of the application of the categories to sensible data structured by the pure forms of intuition.

Of course the synthetic a priori is a principle.

synthetic a priori isn’t a principle, it’s a relation of the content of certain kinds of conceptions to each other; — Mww

Just taking your first point.

Kant writes in CPR B14 section V that a synthetic a priori judgement is a principle of reason.

"Synthetic a priori judgments are contained as principles in all theoretical sciences of reason."

The Merriam Webster Dictionary includes "a comprehensive and fundamental law, doctrine, or assumption" as part of its definition of "principle".

It is true that "synthetic a priori judgements" relate the synthetic with the a priori, but it is also true that a synthetic a priori judgement is a principle that may be used, for example in mathematics (B14)

Kant's synthetic a priori is the principle that relates the synthetic to the a priori.

Kant also refers to other principles, such as the principle of contradiction (B14), principle that all things as appearances are in time (B52), principle of the ideality of our sensible intuitions (B70), principle of the unity of manifold of representations (A117), principle of the unity of apperception (A112), etc.

Over the course of seven days, you’ve included B276 in every single one of seven consecutive responses to my posts to you, but never say any more than the text itself. — Mww

Apart from my ongoing justification, based on the text, that "Kant's synthetic a priori is the principle that we can discover a priori necessity from a posteriori contingency".

One must not overlook the significance embedded in propositions such as, consciousness of determinations of existence in time. — Mww

Kant starts his proof in CPR B276 with "I am conscious of my existence as determined in time", and continues with his proof that "The mere, but empirically determined, consciousness of my own existence proves the existence of objects in space outside me."

I agree with the significance of the passage of time as central to Kant's refutation of Problematic and Dogmatic Idealism.

While that is the case, it is merely beside the point. It needs be shown why external objects as considered by the established idealisms of the day were conceived without proper regard for what came to be posited as transcendental conditions, the foremost being, of course, time. — Mww

In what way is Kant's proof in CPR B276 besides the point, ie "The mere, but empirically determined, consciousness of my own existence proves the existence of objects in space outside me".

Berkeley's Dogmatic Idealism declares that space and all the things in it are impossible and things in space are imaginary (CPR B275)

Kant wanted to show that his idealism is a formal idealism rather than Berkeley's subjective realism, and therefore included in the second edition his Refutation of Idealism (Introduction to CPR).

In the Refutation of Idealism (CPR B276), Kant attempts to prove not only the existence of things outside me, but also that our inner experience of these things outside me is immediate, and we are only conscious of our own existence in time because of the existence of things outside me.

In Practice, it would be difficult fr Berkeleyan Idealism to prove the non-existence of things outside me.

Kant isn’t proving the existence of things as much as he’s proving the material idealist’s denial or doubt of things, is improperly justified. — Mww

In B276, Kant starts with the theorem: "The mere, but empirically determined, consciousness of my own existence proves the existence of objects in space outside me."

He then follows this by his proof: "Therefore it is also necessarily combined with the existence of the things outside me, as the condition of time-determination; i.e., the consciousness of my own existence is at the same time an immediate consciousness of the existence of other things outside me."

In B276, Kant seems to be proving that objects in space and time exist independently of our experience of them, regardless of any belief of the Transcendental Idealist.

But naturalism then presumes that the mind which knows it, is the product of that process it only knows metaphorically. — Wayfarer

That's my question. Are scientific explanations literally true or metaphorically true? Is F = ma literal or metaphorical?

According to Britannica articles:
" naturalism, in philosophy, a theory that relates scientific method to philosophy by affirming that all beings and events in the universe (whatever their inherent character may be) are natural."

"scientific method, mathematical and experimental technique employed in the sciences. More specifically, it is the technique used in the construction and testing of a scientific hypothesis."

"scientific hypothesis, an idea that proposes a tentative explanation about a phenomenon or a narrow set of phenomena observed in the natural world."

Certainly we can imagine the early universe, devoid of organic life, but that imagined universe still contains a perspective and a sense of scale provided by the observing mind. — Wayfarer

It depends on the viewpoint.

We can imagine a sense of scale in an early universe devoid of life from our viewpoint, but we can't imagine a sense of scale in an early universe devoid of life from the viewpoint of an early universe devoid of life.
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Nothing will ever lead me to understand what a nebula, which could not be seen by anyone, might be. Maurice Merleau-Ponty — Wayfarer

It's a question of logic.

Something can be only be named if an observer knows what that something is.

If there is no observer to know what a something is, then it cannot be named.

Therefore, for something to be named "nebula" when there is no one to know what that something is is a logical contradiction.
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We cannot step outside the life-world, because we carry it with us wherever we go. — The Blind Spot - Adam Frank, Marcelo Gleiser, Evan Thompson

The Direct Realist would disagree. I would estimate that 50% of those on the PF believe in Direct Realism.

For the Direct Realist, the world they see around them is the real world itself. Things in the world are perceived immediately or directly rather than inferred on the basis of perceptual evidence.

If a Direct Realist sees an apple on a table, for the Direct Realist, the apple and table exist in the world exactly as they perceive it independently of being perceived.

The Direct Realist does believe that they can step outside the life-world.
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Whereas, I think you're taking what you understand as the scientific picture of the world as being real independently of any observer, attributing with a kind of absolute or taken-for-granted reality. But then you can't see where 'mind' fits in, because that picture is purportedly 'mind-independent'! — Wayfarer

As an Indirect Realist, I agree that I cannot know the early universe independently of my mind, as I can only know the early universe using my mind.

I cannot literally know the early universe, but I can metaphorically know it.

The book Metaphors We Live By 1980 by George Lakoff and Mark Johnson suggests that metaphor is a tool that enables people to use what they know about their direct physical and social experiences to understand more abstract things like work, time, mental activity and feelings.

It can be argued that language, including the language of science, is more metaphorical rather than literal.

Metaphors are commonly used in science, such as: evolution by natural selection, F = ma, the wave theory of light, DNA is the code of life, the genome is the book of life, gravity, dendritic branches, Maxwell's Demon, Schrödinger’s cat, Einstein’s twins, greenhouse gas, the battle against cancer, faith in a hypothesis, the miracle of consciousness, the gift of understanding, the laws of physics, the language of mathematics, deserving an effective mathematics, etc.

For example, Andrew May in Metaphors in Science 2000 makes the point that even Newton's second law is a metaphor, in that it is difficult to see how it could be argued that F = ma "is" the motion of an object in any literal sense.

As I understand gravity using the metaphor of a heavy ball on a sheet of rubber, I understand an early universe metaphorically rather than literally.
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Also see How Time Began with the First Eye Opening — Wayfarer

Those who like mythological interpretations may take the birth of Kronos, the youngest of the Titans, as a symbol of the moment here referred to at which time appears, though, indeed it has no beginning; for with him, since he ate his father, the crude productions of heaven and earth cease, and the races of gods and men appear upon the scene - Schopenhauer

Our understanding is more metaphorical than literal, whether figure of speech, myth as noted by Schopenhauer, analogy or symbolism. After all, language is a symbolic system.

The quoted section is only a synthetic judgement based on a pure a priori intuition. — Mww

CPR B276
Theorem = The mere, but empirically determined, consciousness of my own existence proves the existence of objects in space outside me.
Proof = I am conscious of my existence as determined in time.
All time-determination presupposes something persistent in perception.
This persistent thing, however, cannot be something in me, since my own existence in time can first be determined only through this persistent thing.
Thus the perception of this persistent thing is possible only through a thing outside me and not through the mere representation of a thing outside me.
Consequently, the determination of my existence in time is possible only by means of the existenceb of actual things that I perceive outside myself.
Now consciousness in time is necessarily combined with the consciousness of the possibility of this time-determination:
Therefore it is also necessarily combined with the existence of the things outside me, as the condition of time-determination;
i.e., the consciousness of my own existence is at the same time an immediate consciousness of the existence of other things outside me.

Being conscious of my existence must be prior to any perception of persistence
Kant wrote "I am conscious of my existence as determined in time"

For Kant, we have a priori pure intuitions of time and space. These intuitions are non-empirical, singular, immediate, objective and conscious representations.

The unity of consciousness is central to Kant's transcendental deduction of the Categories, which are a priori pure concepts.

Consciousness of my existence cannot be a consequence of perceiving something persistent, but must be prior to perceiving something persistent.

Therefore being conscious of my existence must be prior to any perception of persistence.
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The quoted part is a pretty good definition of sensation. The assertion as a whole is false, insofar as experience is not of things perceived, but representations of them. — Mww

Kant is trying to prove the validity of Realism
Kant wrote "Consequently, the determination of my existence in time is possible only by means of the existence of actual things that I perceive outside myself."

Kant wrote "actual things that I perceive outside me".

This is the purpose of Kant's Refutation of Idealism, an attempt to prove the existence of objects in space outside a representation of them.
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Maybe YOUR a priori pure intuitions according to your transcendental argument, but if Kant with his means for humanity in general the only two are space and time, and they are the necessary conditions for possible experience, it is the other way around from yours. — Mww

Kant seems to be using a Transcendental Argument
Kant writes
a) I am conscious of my existence as determined in time.
b) All time-determination presupposes something persistent in perception.
c) Thus the perception of this persistent thing is possible only through a thing outside me and not through the mere representation of a thing outside me.
d) Consequently, the determination of my existence in time is possible only by means of the existence of actual things that I perceive outside myself.

In other words
1) My consciousness of existence must be prior to being able to perceive a persistence
2) My perception of a persistence is only possible because of things outside me
3) It is the things outside me that determine the consciousness of my existence

This seems to be a transcendental argument.

What things do we not know through their effects/acts? How could we know anything immanent if not through its effects/acts? — Count Timothy von Icarus

I agree, so let's take this as a starting position.
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The dividing line is at the eye because the mind/brain is assumed to be the dividing line between the world and the observer. Yet one could make the same sort of case for any dividing line one wants to defend. — Count Timothy von Icarus

I agree, in that there have been many different carriers of the information we have in our consciousness about the Andromeda Galaxy. Each change in carrier may be called a dividing line.

One question is, does the change in the carrier of information of necessity require a change in the information carried?

For example, the information about a red star is carried as an electromagnetic wavelength of 700nm through space and an electric current up the optic nerve.

Is it logically possible to maintain exactly the same information when changing from a carrier of oscillating electric and magnetic fields to a carrier of charged particles moving along an electrical conductor?

If it is, the Direct Realists have a persuasive argument that the world we see around us is the real world itself, where things in the world are perceived immediately or directly rather than inferred on the basis of perceptual evidence.

If not, then the Indirect Realists have a persuasive argument that things in the world are perceived indirectly.
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How unknown is it if you know what it causes and that it is red? — Count Timothy von Icarus

I agree when you say: "What things do we not know through their effects/acts? How could we know anything immanent if not through its effects/acts?"

We perceive as a sensory experience the phenomenon that we understand as a red postbox.

We believe that something has caused our sensory experience, in that we don't believe our sensory experience has generated itself.

Regardless of whether we do or don't know the cause of our sensory experiences, we can name the cause "a red postbox".

The name "red postbox" is not the name of a thing-in-itself in a world outside our mind, but rather is the name of the cause of a known sensory experience

In the same way, i) we name the cause of an acrid smell something that is "acrid", as in "acrid smoke from a bonfire", ii) we name the cause of a bitter taste something that is bitter, as in "angostine bitters", iii) we name the cause of a sweet taste as something that is sweet, as in "a sweet apple", we name the cause of a silky feel as something that is silky, as in "silk" and finally v) we name the cause of crackling sound as something that crackles, as in "a cracking fire".

IE, we can name the cause of our sensory experiences after the known sensory experience, meaning that we can even talk about an unknown cause as "the cracking fire giving off acrid smoke".
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This seems to be equivocating between different sorts of mind-independence. — Count Timothy von Icarus

In one sense, in philosophy-speak, "the early universe" is not mind-independent, in that "the early universe" are words which only exist in language, and language only exists in the mind. As "the early universe" only exists in the mind, by definition, it cannot be mind-independent.

In another sense, in ordinary language, "the early universe" was clearly mind-independent, and can be affirmed by every non-philosopher, by every person going to the supermarket or waiting at a bus stop,

For those philosophers that do believe that "the early universe" is not mind-independent, then logically no referent of any expression can be mind-independent, including expressions such as "tables and chairs". This which would be a problem for Direct Realists, as they do believe that things like tables and chairs do exist independently of the mind, that they are perceived immediately or directly rather than inferred on the basis of perceptual evidence.

That appearances are the necessary antecedent occasions for their employment, it does not follow they are derived from them, and in accordance with the theory, they are indeed, not, nor can they be. — Mww

Kant writes in CPR B276 on the Refutation of Idealism: "I am conscious of my existence as determined in time............................Consequently, the determination of my existence in time is possible only by means of the existence of actual things that I perceive outside myself."

"I am conscious of my existence as determined in time" is an a priori pure intuition.

"The existence of actual things that I perceive outside myself" is a posteriori empirical experience.

Therefore "Consequently, the determination of my existence in time is possible only by means of the existence of actual things that I perceive outside myself" can be read to mean that my a priori pure intuition is possible only by means of a posteriori empirical experience.

This is a transcendental argument.

I'd rather say the relationship between some red object and someone seeing that object as red is essentially of the same sort that exists between two meteors colliding in interstellar space — Count Timothy von Icarus

I perceive as a phenomenal experience a red object, and believe that there is some unknown thing the other side of my senses that has caused this phenomenal experience. For convenience, I call this unknown thing a red object.

As two meteors collide, something the other side of my phenomenal experiences has caused my phenomenal experiences.
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Yet causation, information, energy, etc. seem to flow across the boundaries of animal bodies as if there was no boundary at all, so I see no reason to presuppose such a dividing line. — Count Timothy von Icarus

I agree that information flows across these boundaries, but would add that the carrier of the information changes across such boundaries, meaning that there is a dividing line.

For example, on the other side of the eye, the carrier of the information is a wavelength of 700nm travelling through space, and on this side of the eye the carrier of the information is an electric signal travelling up the optic nerve.
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Well, presumably the number 700 doesn't exist outside minds either, right? — Count Timothy von Icarus

Exactly.
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How do we reconcile this seeming multiplicity (the Many) with the equally apparent unity of being (the One)? — Count Timothy von Icarus

We perceive complex patterns, whether inorganic, such as the rhombic dodecahedral crystal of a garnet, or organic, such as the stripes on a tiger.

But within such patterns we perceive a unity. Kant calls this "transcendental apperception". But how "transcendental apperception" is possible is beyond my understanding. How can the mind be conscious of a unity outside of time and space (the One) when the parts are separated in time and space (the Many)?
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But thought is obviously something with being. — Count Timothy von Icarus

Yes, we cannot think about "being" without thoughts, but we can also think about "being" existing outside the mind. Otherwise we come to the conclusion that the Universe didn't begin 13 billion years ago, but only began 200,000 years ago when humans developed language.
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I don't think solipsism is good philosophy.....................My take would be that we experience the things we do for reasons, due to causes, etc. and such reasons do not bottom out in the inaccessible and unintelligible as soon as we leave the confines of our own discrete phenomenological horizon. — Count Timothy von Icarus

I agree, because of my belief beyond doubt that some of my phenomenal experiences have been caused by something the other side of such phenomenal experiences.

My belief is that my belief beyond doubt in causation is a consequence of life having evolved for 3 billion years in dynamic interaction with its environment, ie Enactivism, resulting in my belief in causation having become part of the physical structure of my brain, ie Innatism.

I think you're wrestling with a real conundrum inherent in modern culture and philosophy. — Wayfarer

When driving in a city and see a red traffic light, I know what to do, which is to stop the car.

Do I need to know more than what I have directly perceived, a red traffic light, or do I need to worry whether a prior cause was a wavelength of 700nm?
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Kant maintains that the structures of cognition, like time and space, are necessary preconditions that shape any experience we might have; and that they are not derived from or contingent upon empirical experiences. — Wayfarer

The relevance of Kant
The OP asks "Can thought explain being?..................What we really want is an explanatory structure that preserves both of the seemingly ineluctable realities – of logic and of being."

Kant's synthetic a priori in the CPR seems appropriate to the OP, where the synthetic is about knowing the being of a world outside the mind and the a priori is about thought and logic inside the mind.

The relationship between a priori logical necessity and a posteriori empirical experience
Kant in CPR A2 discusses the transcendental nature of the relationship between a priori inner necessity and empirical experience.

Kant in the CPR proposes that we have a priori pure intuitions of space and time and a priori pure concepts of the Categories.

As you say: "The a priori nature of space and time is fundamental to Kant’s project, establishing these as the conditions that make empirical knowledge possible in the first place."

One question is, how does Kant explain the origin of these a priori pure intuitions and a priori pure concepts?

There are three possibilities. Either i) we are born with them, or ii) from self-causation or iii) from empirical observation.

As regards i), we know that from birth babies have an awareness of the pure intuitions of space and time and the pure concepts of the Categories, yet Kant was critical of the Rationalist version of Innateness (SEP - The Historical Controversies Surrounding Innateness).

As regards ii), as Causation is a Category, causal explanations rather than non-causal explanations are part of the CPR.

As regards iii) there are two reasons to support this.

First, Kant was aware of Newton and necessary mathematical laws derived from empirical observation. As you say "Modern mathematical physics is full of examples where mathematical reasoning anticipates empirical confirmation, from the countless 'Einstein proved right, again!' headlines to Paul Dirac's prediction of anti-matter, which, as he famously said, 'fell out of the equations.'"

Second, from the text of B276, the Refutation of Idealism, where Kant writes:
i) I am conscious of my existence as determined in time.
ii) Consequently, the determination of my existence in time is possible only by means of the existence of actual things that I perceive outside myself.
IE, the a priori "I am conscious of my existence as determined in time" is a consequence of the a posteriori "existence of actual things that I perceive outside myself.", and is a transcendental argument.

For the above reasons, it seems to me that the CPR only makes sense if a priori necessity has transcendentally derived from a posteriori contingency.

Where in the pertinent text might I find support for such an assertion? — Mww

Synthetic a priori knowledge is central to the thought of Immanuel Kant, who argued that some such a priori concepts are presupposed by the very possibility of experience (Britannica - synthetic a priori proposition)

In CPR A2, Kant starts by explaining transcendental philosophy. He separates a priori cognitions, universal, independent of experience and having an inner necessity from a posteriori cognitions, dependent upon empirical experience.

Therefore, there is a priori necessity and a posteriori contingency.

However Kant is not an Innatist, in that a priori necessity is not something we are born with. He uses a transcendental argument that although cognition of inner necessity is prior to a posteriori empirical cognition, such a prior cognition has in fact been determined by a posteriori cognition.

Kant gives an example of a transcendental argument in CPR B276 in his Refutation of Idealism. For example, my consciousness of my existence in time depends on perceiving an actual thing outside me, which depends on my consciousness of my existence in time.

For Kant, a prior necessity can be transcendently deduced from a posteriori contingency.

The term does invite confusion as it stands. — J

:up:

Enactivism, by contrast, is focused on dissolving the strong subject-object dualism that is presupposed by the division of thought from being. — Count Timothy von Icarus

I agree that life evolved through a dynamic interaction between itself and its environment over billions of years, such that life is an intrinsic part of of its whole environment.

There is thought and being.

The question is, how is it logically possible to overcome the dualism between thought and being when life only knows about being through thought?

If all the contents of experience cannot be said to "exist in the world" in virtue of "only existing in the mind," I don't see how that isn't denigrating the relationships that exist between things and thinking beings as in a way "less than fully real." — Count Timothy von Icarus

I know that I perceive the colour red and feel pain.

I believe that neither the colour red nor pain exist in the world.

I believe that sometimes my perceptions of red and pain have been caused by something this side of my senses, such as dreams and headaches, and sometimes have been caused by something the other side of my senses, such as the wavelength of 700nm or a thistle.

As I don't believe that pain exists in the thistle, I don't believe that the colour red exists in the wavelength of 700nm.

I believe that the cause of my perceptions is as real as the perceptions themselves, though not necessarily the same, in that the perception of pain is not the same thing as its cause, a thistle.

Do you believe that the colour red and pain exist in a world outside a mind?
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If we're allowing the world to be unintelligible and unknowable why not simply allow that Y (the mind) generates itself as a brute fact? — Count Timothy von Icarus

If that were the case, that there is no world the other side of my senses, and my mind has generated itself, then that means I wrote "War and Peace", composed Symphony No. 9 from "The New World" and developed the special and general theories of relativity.

Of the two hypotheses, that someone else wrote "War and Peace" or I wrote it, the simplest explanation is that someone else wrote it.

Do you think that Idealism is a simpler solution than Realism?

It's this relationship between mathematical logic (DME) and contingent causation that is central to the argument. — Wayfarer

As an Indirect Realist, I agree with much of what you say.

I perceive the colour red even though I believe that the colour red doesn't exist in the world. I perceive pain even though I believe that pain doesn't exist in the world. I perceive numbers even though I believe that numbers don't exist in the world.

@Wayfarer: Mathematics doesn’t require numbers to exist as physical objects.
Numbers, therefore, need not exist in the world to guide explanations of physical forces, provided they symbolically represent the appropriate values.
I think the 'practical problem' you're referring to, is how numbers can be real if they don't exist in a physical sense.

However, even though I believe that the colour red, pain and numbers don't exist in the world, I believe there is something real in the world that has caused my perception of the colour red, pain and numbers, even though I will probably never know what it is.

@Wayfarer: The fact that mathematical reasoning often anticipates empirical phenomena (such as Dirac’s prediction of anti-matter) suggests a deep correspondence between mathematical structures and causal relations in the world.

As I know that my perceptions are real, I believe that the cause of my perceptions are also real, even if I will never know what these causes are.

@Wayfarer: Whatever mathematical system we invent must, by necessity, align with these constraints to be applicable.

The fact that we invent maths does not mean that it is arbitrary, in that it is only useful to us if it corresponds with what we observe.

@Wayfarer: You can't get around it by declaring that mathematics is purely arbitrary, because it ain't.

What does "Contingent causal law" mean?

@J: whether “the facts under question arise from a degree of mathematical necessity considered stronger than that of contingent causal laws.”

There is a difference between the words "contingent" and "nomic".
A "contingent law" would be: "take your shoes off when entering a house in Japan"
A "nomic law" would be: "All bodies attract each other with a force inversely proportional to the square of the distance between them"
Isn't "contingent causal law" a contradiction in terms?

Mathematical logic and contingent causation

I invent Maths A whereby 1+1=3 and subsequently discover that it doesn't correspond with what I observe, so I discard it.
Within the mathematical logic of Maths A, 1=1 is necessarily 3, but doesn't agree with contingent observations.

I then invent Maths B whereby 1+1=2 and subsequently discover that it does correspond with what I observe, so I keep it.
Within the mathematical logic of Maths B, 1=1 is necessarily 2, and does agree with contingent observations.

Even if Maths B does agree with contingent observations, it doesn't logically follow that Maths B is necessarily true, because there is no guarantee that a particular observation will be discovered that it doesn't agree with.

Well, enactivism is generally presented as a counter to indirect realism and representationalism. — Count Timothy von Icarus

Enactivism is no counter to Indirect Realism
It would seem that the physical structure of the brain is a consequence of around 3 billion years of evolution, shaped by a dynamic interaction between life and its environment (Enactivism)

Such a physical brain may well be born with "knowing how", such as how to feel pain, how to see the colour red, but not "knowing that", such as the sky is blue, snow is cold. See Gilbert Ryle and his book The Concept of Mind (Innatism)

Even accepting Enactivism and Innatism, it remains true that during its life, this physical brain can only gain new information about any outside world through its five senses, through its sensory experiences (I have seen no evidence for telepathy).

The problem remains, sensory experiences remain representations of what exists the other side of he senses.
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It seems to me that the way we get into trouble here is by positing knowledge of things "in-themselves" as the gold standard of knowledge — Count Timothy von Icarus

As an Indirect Realist, I agree.

When driving through a city, all I need to know is whether the traffic light is red or green. That the thing-in-itself is emitting a wavelength of 700nm or 500nm is of little immediate import.
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I suppose another related issue lies in correspondence theories of truth. One can never "step outside experience," in order to confirm that one's experiences "map" to reality. But this to me simply seems to suggest something defective in the correspondence theory of truth. — Count Timothy von Icarus

The Correspondence Theory of Truth is a problem for the Indirect Realist, but not for the Direct Realist, who believes that things in the world are perceived immediately or directly rather than inferred on the basis of perceptual evidence.
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One of the claims that is often made by the representationalist position that Sokolowski critiques is that many of the properties of objects that we are aware of do not exist "in-themselves," and are thus less than fully real

I, as an Indirect Realist, don't see it that way.

Indirect Realism, aka Representationalism, holds the position that any world the other side of the senses is fully real. This is why it is called "Realism". However, what we perceive is only a representation of what exists in any world.

The properties we perceive are representations of the properties that exist in the world.

This doesn't mean that the properties in the world as less real, because if they were, we wouldn't have had any perceptions in the first place.

The mind perceives fully real properties, believed to have been caused by fully real properties in the world, which may or may not be the same as what we perceive.

Arithmetic and mathematical reasoning exemplify this because they allow us to grasp necessary truths that, although not sensory, still inform our understanding of the world — Wayfarer

Is 1 + 1 = 2 a necessary truth by definition or because in the world 1 + 1 = 2?

If I invent a mathematics and define 1 + 1 = 3, then within my mathematics 1 + 1 = 3 is a necessary truth.

If in the world 1 + 1 = 2, then in mathematics 1 + 1 = 2 would be a necessary truth. However, this depends on justifying that numbers exist in the world.

If numbers did exist in the world, then this would require a relation between 1 and 1. But what has not been shown is the ontological existence of relations in the world.

The ontological existence of relations in the world introduces a number of practical problems, suggesting that numbers don't exist in the world.
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I'm simply drawing an analogy to show how there are forms of knowledge, like mathematical deduction, that function beyond sensory input and can help us conceive of Kant’s transcendental structures. Modern mathematical physics is full of examples where mathematical reasoning anticipates empirical confirmation — Wayfarer

Deduction is important in being able to make decisions. For example, i) every day the sun rises in the east, ii) tomorrow will be a day, iii) therefore, tomorrow the sun will rise in the east. Reasoning can anticipate empirical confirmation.

However, deduction cannot change a belief that "every day the rises rises in the east" into knowledge that "every day the sun rises in the east".

In other words, belief cannot transcend into knowledge by reason alone.

Kant's synthetic a priori is the principle that we can discover a priori necessity from a posteriori contingency

Even Kant never justified this, perhaps because it can never be justified.

Well, a question here is what it means to be "independent from observers. — Count Timothy von Icarus

How can an observer observe something that cannot be observed

As an Indirect Realist, I directly know my sensory experiences of sight, sound, touch, taste and smell.

I have an instinctive belief that these sensory experiences have been caused by something rather than being self-generating.

I reason that some causes are this side of my sensory experiences, such as dreams and hallucinations, and some causes are the other side of my sensory experiences, which can be called the mind-independent world.

My belief is in Enactivism, in that life has evolved for about 3 billion years through a dynamic interaction between an organism and its environment.

One can sensibly reason that certain knowledge essential for survival in this environment, such as a belief in causation, have become an integral part of the physical structure of the brain, meaning that my belief in causation is beyond doubt.

I don't know that there is a mind-independent world, but my belief in such a world is beyond doubt.

I know a set of consistent sensory experiences, such as green in colour, without sound, smooth in touch, sweet in taste and slightly acrid in smell and name this set of consistent sensory experiences "apple".

"Apple" is not the name of something in a mind-independent world, but rather the name of a consistent set of sensory experiences.

In answer to the question, how can an observer observe something that cannot be observed, the answer is that they cannot

However, an observer can observe their own sensory experiences, which they can reason have been caused by an unknown something the other side of these senses, which can be called a "mind-independent world".

In other words, the "mind-independent word" is not the name of an unknown thing, but rather is the name for an unknown cause of known sensory experiences.
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I would say the weight of virtually all empirical evidence is that an apple being an apple doesn't depend on us specifically for its existence. When we leave a room, the apples don't vanish. — Count Timothy von Icarus

This takes me back to my previous question.

An observer can see that atom A (metaphorically speaking) has a direct relationship with apple X, and atom B has a direct relationship with apple Y.

But when the observer leaves the room, what is the explanation that the atoms have maintained these particular relationships?

We can ‘see’ things through deductive inference that are not empirically knowable. There’s a sense in which even arithmetic is transcendental in that it reveals aspects of nature which sense could not otherwise discern. — Wayfarer

The problem with transcendental arithmetic.

For example, using deductive inference it is possible to prove that the sum of every two integers is always even, something that is not provable empirically.

Deductive inference requires strong axioms and logic. In the above example, one axiom is that a + b = b + a.

However the axiom has been determined prior to any deductive inference, meaning that any result of the deductive inference depends on the axioms chosen. If a different axiom had been chosen, a different result would have been deductively inferred.

A transcendental situation is where not only i) has the axiom been determined prior to the result of the deductive inference but also ii) the axiom has been determined by the result of the deductive inference.

How would this be logically possible?

So we shouldn't limit a robust correspondence theory to "facts about the world" — J

The question is, where is this "world", inside or outside the mind.

The relationship between the Correspondence Theory and Truth is an enormous subject, as indicated by the SEP articles on Truth and The Correspondence Theory. The Merriam Webster Dictionary has a range of definitions of "truth", and the word "truth" may be used both literally and as a figure of speech.

Even if we adopt the schema "snow is white" is true IFF snow is white, which relates something in language to something in the world, the next question is where is this world. The answer depends on individual opinion, whether one is a metaphysical Realist or Metaphysical Idealist, never mind all the sub-divisions of these two positions.

For example, are there infinitely many prime numbers.

Initially, we can use the Schema and say that the proposition "there are infinitely many prime numbers" is true IFF there are infinitely many prime numbers.

However, Euclid has proved that there are.

Therefore, we can use the Correspondence Theory and say that the proposition "there are infinitely many prime numbers" is true because there are infinitely many prime numbers.

The proposition "there are infinitely many prime numbers" exists in language.

The next question is, where do these infinitely many prime numbers exist.

In particular, where does the number 23 exist? Some say only inside the mind and some say both inside and outside the mind.

However, regardless of where the prime numbers exist, the Correspondence Theory is still valid.

That is, how can being be in one sense "one," i.e. everything interacts with everything else, there are no truly isolated systems — Count Timothy von Icarus

Yes, Aristotle considered the Platonic argument that he called "The One Over The Many", whose premiss might be expressed as "Whenever two or more things can be properly said to be F, it is by virtue of some thing, F-ness, that they are properly called F"

In the world are observed numerous atoms (philosophically speaking).

For the observer, atom A and atom B are part of one apple and atom C is part of a different apple.

For me, the problem about objects existing in a world independent of any observer, is, what mechanism is there in such a world independent of any observer that relates atom A to atom B but doesn't relate atom A to atom C?

1) So, the neo-logos philosophies might say something like, "If nature has patterns, and our language has patterns, and we are derived from nature, it may be the case that our language is a necessary outcome of a more foundational logic". Thus, the logic would not be transcendental, but (for lack of a better term) "immanent" in nature, not some outside observing entity that is detached from it.
2) Kant never explains why our minds would compose such a world, but evolution does.
3) Wouldn't evolution put a connection between the efficacy of the mind and the world?
4) But what is the world outside of an observer?
5) Otherwise it's just "I have believe" without an explanation, which though is valid in terms of asserting an idea, is not necessarily valid as an fully informed reason for why you think that way. — schopenhauer1

1) 2) 3) I agree in the sense that I believe in Enactivism, where life has evolved over 3 billion years in a dynamic interaction with its environment, meaning that life is a part of the world, not outside it.

2) Yes, Kant argues in the CPR that we have transcendental knowledge of a world the other side of our senses, but never explains how this could work.

4) We are an intrinsic part of the world as the tree is an intrinsic part of the world, but it doesn't follow that because we are an intrinsic part of the world we must necessarily have knowledge about the world that we are an intrinsic part of.

The problem remains that our only knowledge about any world the other side of our senses (sight, sound, taste, touch, smell) arrives through these very same senses.

5) For myself, as an Indirect Realist, I believe that there is a world the other side of my senses that has caused what I experience in my senses, and can justify my belief in such a world.

The problem remains, we may reason about what has caused the experiences in our senses, and arrive at beliefs about any world the other side of our senses, but how can we transcend our senses in order to know what has caused these experiences in our senses?

I know the colour red from my sense of sight. I may have a reasoned belief as to its cause, but how can I ever know its cause?

Even Enactivism (that life has evolved as an intrinsic part of the world) provides no mechanism whereby we can know what exists the other side of our senses.

This is Jha et al’s argument, more or less. Math only appears to be causal when we state the problem in terms that remove, or demote to “background conditions,” the physical constraints that actually provide the explanation. — J

This might be a possibility.

Is 1+1=2 a mathematical necessity in the world?

It initially seems that nothing in the world has caused 1+1 to be 2, supporting Lange's DME that mathematical necessity is stronger than contingent causation.

Where P = 1+1. The antecedent, the prior situation in the world.
Where R = 1+1=2. The explanans, the necessary mathematical law.
Where Q = 2. The consequent, the subsequent situation in the world.

Let P be one apple alongside apple, in other words 1+1
Let Q be a pair of apples, in other words 2.

But for there to be a pair of apples, this presupposes a prior relation between the individual apples.

In other words, the existence of a pair of apples has been caused by a prior relation between the individual apples.

Therefore, the mathematical law that 1+1=2 is a consequence of a prior relation between 1 and 1.

The prior relation between parts is the cause of the necessity of the mathematical law that 1+1=2.

(However, a subsequent question is, do relations ontologically exist in the world?)

So what I think needs to be questioned is why we think a hard and fast separation can be made between mind and world in the first place. — Wayfarer

:100:
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I think the sentiment against this relationship goes back to the same basically nominalist and empiricist attitude that animates most analytical philosophy, which is unwilling to admit that ideas - mathematical ideas, logical principles - might possess any kind of intrinsic reality which can't be reduced to 'contingent causal laws'. — Wayfarer

In our mind we have mathematical ideas such that ${d = 0.5 * g * t^{2}}$ (distance with time of a falling object under gravity), and logical principles such as the Law of Non Contradiction whereby I cannot feel pain and not feel pain at the same time.

If mathematical laws cannot be reduced to contingent causal laws, what caused d to be ${0.5 * g * t^{2}}$ rather than ${3.0 * g * t^{2}}$, for example?

If logical principles cannot be reduced to contingent causal laws, what caused my inability to feel pain and not to feel pain at the same time, for example?

That's a very creative image. But I don't see TS and Madonna in it at all. — Corvus

Can't you see Madonna in the eyes and a nose strikingly similar to that of Taylor Swift?

Yes, we are both perceiving an object that doesn't exist.

If a correspondence theory of truth demands that we do so, I'd argue that it represents a reductio ad absurdum and should be rejected on that ground. — J

Very harsh. The word "true" has different meanings in different contexts.

As regards the equation ${d = 0.5 * g * t^{2}}$, i) I can say that it is true, meaning that it works, ii) I can say that it is true, meaning that it has the correct structure of an equation, iii) I can say that it is true, meaning that it is important to me, as in a true friend, iv) I can say we don't know whether it is true or not, as we don't know what obtains in the world, v) I can say I don't know if it is true or not, as I don't know what the symbols represent.

The scientist can use meaning i) for "truth" and the philosopher can use meaning iv) for "truth".
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What would be the point of limiting ourselves in this way? — J

Yes, the correspondence theory does not tell us about things like pain, which, I believe, doesn't exist in the world.

In the proposition "it is true that I feel pain", what purpose does the word "true" have, in that the proposition says no more than "I feel pain"

It seems that "truth" only arises in the correspondence between the mind and the world.

So, without observers, nothing is related to anything whatever. That is the thrust of the OP — Wayfarer

It seems to me that the OP is about how the mind explains the world.

A sub-topic is about does the mind explain the world using mathematical necessity or causal contingency.
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As per above, this question can be asked not only of patterns, but of phenomena generally..........................There are patterns that appear in inorganic nature, in crystals, snowflakes, larva formations etc. — Wayfarer

I agree with the introduction to Pinter's book Mind and Cosmic Order

The book’s argument begins with the British empiricists who raised our awareness of the fact that we have no direct contact with physical reality, but it is the mind that constructs the form and features of objects. It is shown that modern cognitive science brings this insight a step further by suggesting that shape and structure are not internal to objects, but arise in the observer.

I read this as saying that patterns exist in the mind, not the world.

Is this meant to be Tarski's view? Surely he didn't talk about what was the case in the world -- only about the correct relations between language and metalanguage. If one language has to be "about the world," then we wouldn't have any logical or mathematical truths at all, or at least that seems to be the necessary consequence. I don't think Tarski intended this. Unless I'm mistaken, he included these kinds of truths in his schema. — J

I think of it as more the Correspondence Theory of Truth, in that a belief is true if there exists an appropriate entity, a fact, to which it corresponds. (SEP - Truth)

I agree that such a Correspondence Theory of Truth draws on ideas developed by Tarski, who was more concerned with mathematical logic than the metaphysics of truth (SEP - Truth).

Tarski's Semantic Theory of Truth (STT) is considered to be a version of Aristotle's Correspondence Theory of Truth, and treats truth as relative, rather than the classical approach of treating truth as absolute.

Considering the schema "snow is white" is true IFF snow is white, Tarski's STT only applies to formal languages, in that "snow is white" is within the object language whilst snow is white is within the metalanguage. Within the modern correspondence theory, "snow is white" is also within the object language whilst snow is white is a fact in the world.

It may well be that we don't know whether an equation is mathematically true or not, but pragmatically, does this matter as long as the equation works. All a scientist wants to know is that an equation works. Even if the scientist did know that the equation was a mathematical truth, this wouldn't affect their use of the equation.

An object in the world emits a wavelength of 700nm and I perceive a red light. I am driving a car, see a red light on a traffic light and know to stop. Have I stopped because I know the truth , that a wavelength of 700nm has been emitted from the traffic light, or have I stopped because I see a red light?

But, all organic life displays just the kind of functional unity that a painting does, spontaneously. Those patterns most definitely inhere in the organic world. DNA, for instance. — Wayfarer

Can the thought of a pattern in the mind explain the being of a pattern in the world

It is true that we can see many examples of patterns which we find have an aesthetic beauty, and which have arisen spontaneously because of the laws of nature. The web site Natural Form Patterns shows many examples.

We see patterns in the world, but the question is, do the patterns that we see exist in the world or only in our mind. Did patterns exist in the world prior to there being anyone to see them?

A pattern has a unity because of the particular way things are spatially and temporally related to each other, where the whole is more than the sum of its parts.

For example, we can see spatial relationships between the blobs of paint on a Monet canvas creating an aesthetic unity in our minds. However, in the absence of an observer, what is the ontological nature of the spatial relationship between these blobs of paint?

Do spatial and temporal relations ontologically exist in a world absent of any observer?

Because if within a world absent of observers, spatial and temporal relationships had no ontologically existence, then there would be no way of spatially and temporally relating disparate things together, meaning that in the world there would be no patterns, as a pattern can only exist if its parts are somehow related together.

For example, two Monet paintings are alongside each other, "St Lazare Station" and "Water-lilies". We, as observers, can see that there are two different paintings, where the blobs of paint in "St Lazare Station" make one unified whole, and the blobs of paint in "Water-lilies" make a different unified whole.

But in the absence of any observer, what mechanism exists in the world that relates one blob of paint in "St Lazare Station" to another blob of paint in "St Lazare Station" but not to another blob of paint in "Water-lilies"?

In the absence of any observer, how can thing A relate to thing B but not to thing C?

Because if either i) thing A neither related to thing B nor thing C or ii) thing A related to both thing B and thing C, then there would be no patterns existent in a world absent of any observer.

I understand. Do you think there are mathematical truths that are independent of what is the case in the world? Plain old theorems, in other words? — J

Taking an example. Do I think that the theorem "angles on one side of a straight line always add to 180°" is true independent of what is the case in the world.

What is truth?

My understanding of truth is that it is defined by the schema "snow is white" is true IFF snow is white, where "truth" is the correspondence between propositions in language and equations in mathematics and what is the case in the world.

If I am correct, then if a proposition in language or an equation in mathematics is independent of what is the case in the world, then by the definition of truth, such a proposition or equation can neither be true nor false.

A proposition may work, such as "the sun rises in the east" and an equation may work, such as "1 + 1 = 2", but the fact that they work doesn't mean that they are true, if truth is defined as a correspondence between something in the mind and something in the world.

The problem is knowing what is the case in the world.

I have found the SEP article on Mathematical Explanation, which I haven't read yet, but it should make clearer Lange's idea of dividing 23 strawberries equally amongst three friends.

So I assumed you regarded d=0.5∗g∗t2 as a mathematical truth. — J

I would agree that the equation ${d = 0.5 * g * t^{2}}$ is a mathematical truth IFF ${d = 0.5 * g * t^{2}}$ is the case in the world.

However, who knows what is the case in the world?

Who do you get if you amalgamate images of Elon Musk with Bill Gates.................................Why would you do that? — Corvus

To scare children on Halloween!?



https://openart.ai/discovery/sd-1006000370197221428

"Are the equations being imposed or simply reflected in the mathematics?" — schopenhauer1

Both. For 100 days we observe the sun rise in the east, and invent the rule "the sun rises in the east". The rule reflects past observations, but is no guarantee that the rule will still apply in the future. We impose the rule on the world, in the expectation that the rule will still apply in the future.
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Some neo-Logos philosophies might say the mind cannot but help seeing the very patterns that shape itself. — schopenhauer1

I'm with Kant on that.
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I can imagine a type of pattern whereby the mind works (X), and a pattern whereby the world works Y, and X may be caused by Y, but X is not the same as Y. — schopenhauer1

Exactly. A postbox emits a wavelength of 700nm ( Y) which travels to the eye which we perceive as the colour red (X), where our perceiving the colour red in the mind was caused by the wavelength of 700nm in the world.

There is the general principle that an effect may be different in kind to its cause. For example, the effect of a pane of glass breaking is different in kind to its cause of being hit by a stone.
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Is our language contingently relating with the world or necessarily relating to the world. — schopenhauer1

Perhaps the difference is temporal. Going forwards in time, from cause to effect, the pane of glass of necessity breaks when hit by the stone. Going backwards in time, from effect to cause, the breaking of the glass was contingent on being hit by a stone, but equally it could have been hit by a bird.
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I can see a sort of holistic beauty in the aesthetic of the language reflecting the world because it is derived from (the patterns) of the world. — schopenhauer1

Aesthetics is perceiving a unity in the whole from a set of disparate parts. For example, the magic of a Monet derives from the artist's deliberate attempt to create a unity out of a set of spatially separate blobs of paint on a canvas. Such a unity exists only in the mind of the observer, not in the world, in that one blob of paint of the canvas has no "knowledge" as to the existence of any other blob of paint on the canvas. Patterns only exist in the mind, not the world.

As patterns don't ontology exist in the world, but do exist in the mind, to say that patterns in the mind have derived from patterns in the world is a figure of speech rather than the literal truth.
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I can see a sort of nihilistic "contingency" in the aesthetic of language never really derived from, but only loosely reflecting the world. — schopenhauer1

Perhaps it is more the case that the aesthetic brings meaning out of the meaninglessness of nihilism. It is the aesthetic that discovers the unity of a whole within disparate parts, finds patterns in randomness and seeks sense out of senselessness. For example, the aesthetic of Picasso's Guernica shows us the possibility of a greater good born out of the nihilism of war, and the aesthetic of the mathematical equation shows us a greater understanding born out of a nihilistic Universe that is fundamentally isolated in time and space.

Are you sure we should call something like d=0.5∗g∗t2 a mathematical truth? I thought it was only true on some interpretation; as it stands, it has no meaning. — J

${d = 0.5 * g * t^{2}}$ is a mathematical equation and it works. It is true that it works, but that does not mean it is a mathematical truth.

@RussellA: Eventually, after many attempts, we invent the equation ${d = 0.5 * g * t^{2}}$, discover that it works, and keep it.................We know it works, but we don't know if it is a necessary truth.

What do you mean that the equation ${d = 0.5 * g * t^{2}}$ has no meaning?

Gravity, electromagnetism, chemical interactions, biological interactions, etc. work ways that impose on us their workings, not the other way around........................Kant, for example, seemed to conflate the two as part of the same "transcendental" constraints that our minds impose on "the thing-itself". — schopenhauer1

Does the world impose itself on the mind or does the mind impose itself on the world?

Perhaps its a combination of both.

We observe regularities in the world.

We then invent the equation ${d = 1.0 * g * t^{2}}$, discover that it doesn't work, and discard it. We then invent the equation ${d = 2.0 * g * t^{2}}$, again discover that it doesn't work, and discard it. Eventually, after many attempts, we invent the equation ${d = 0.5 * g * t^{2}}$, discover that it works, and keep it. In one sense, the world has imposed itself on us, in that the world has "determined" that the equation ${d = 0.5 * g * t^{2}}$ works, not us.

However, in another sense, we impose the equation ${d = 0.5 * g * t^{2}}$ onto the world, in that following Hume's concept of knowledge by constant conjunction, any correspondence between the equation and the world may be accidental. Today the equation may work, but tomorrow it may not. We only know in a pragmatic sense that the equation does work. We don't know why it works. Because we don't know why the equation works, we are not able to say that it will always work, as the equation doesn't contain within itself its own proof.

The world imposes itself on us which equation we use, but we impose our equation onto the world, even though the equation may not correspond with any underlying reality within the world.

Are mathematical truths necessary truths
Following the schema "snow is white" is true IFF snow is white as a definition of "truth", then "
${d = 0.5 * g * t^{2}}$" is true IFF ${d = 0.5 * g * t^{2}}$

But the mathematical equation "${d = 0.5 * g * t^{2}}$" has originated from observations of constant conjunctions within the world (using Hume's term), it hasn't originated from a proof derived from a knowledge of the fundamental reality of the world.

Therefore, we don't know if it is the case that in the world ${d = 0.5 * g * t^{2}}$. It then follows that we don't know if "${d = 0.5 * g * t^{2}}$" is true. We know it works, but we don't know if it is a necessary truth.

Kant and a Transcendental Deduction that mathematical truths are necessary truths

In B276 of the CPR, Kant uses a Transcendental Deduction to prove the existence of objects in the world.

As the equation "${d = 0.5 * g * t^{2}}$" does successfully and consistently predict what is observed in the world, we could use a similar Transcendental Deduction to prove that in the world is the underlying reality that ${d = 0.5 * g * t^{2}}$.

Using such a Transcendental Deduction, we could unify a world that imposes itself on the mind and a mind that imposes itself on the world.