B xvi-xv, ¶ 9-10

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[¶9] Metaphysics - a wholly isolated speculative cognition of reason that elevates itself entirely above all instruction from experience, and that through mere concepts (not, like mathematics, through the application of concepts to intuition), where reason thus is supposed to be its own pupil - has up to now not been so favored by fate as to have been able to enter upon the secure course of a science, even though it is older than all other sciences, and would remain even if all the others were swallowed up by an all-consuming barbarism. For in it reason continuously gets stuck, even when it claims a priori insight (as it pretends) into those laws confirmed by the commonest experience. In metaphysics we have to retrace our path countless times, because we find that it does not lead where we want to go, and it is so far from reaching unanimity in the assertions of its adherents that it is rather a battlefield, and indeed one that appears to be especially determined for testing one's powers in mock combat; on this battlefield no combatant has ever gained the least bit of ground, nor has any been able to base any lasting possession on his victory. Hence there is no doubt that up to now the procedure of metaphysics has been a mere groping, and what is the worst, a groping among mere concepts.

[¶10] Now why is it that here the secure path of science still could not be found? Is it perhaps impossible? Why then has nature afflicted our reason with the restless striving for such a path, as if it were one of reason's most important occupations? Still more, how little cause have we to place trust in our reason if in one of the most important parts of our desire for knowledge it does not merely forsake us but even entices us with delusions and in the end betrays us! Or if the path has merely eluded us so far, what indications may we use that might lead us to hope that in renewed attempts we will be luckier than those who have gone before us?

Summary

Metaphysics has failed to become a secure science. It has constantly had to begin over again so that no advance has been made. However, if metaphysics were impossible, then we must wonder: why does our own nature seem to aim us in the direction of these problems?

Commentary

In this passage, metaphysics is both ridiculed and honored. On the one hand, metaphysics has failed as a science and is only suited for mock combat. Perhaps worst of all it is a science that deals merely in concepts that - unlike natural science or mathematics - cannot be exhibited anywhere: metaphysics seems to be a game for thought. On the other hand, metaphysics concerns problems that are of the highest moment for us, and so metaphysics is the oldest and most enduring science; it seems a product of our own nature calling out for answers. From this situation Kant faces a crisis: if metaphysics is really a dead end, then what can we say for ourselves since our very nature leads us down this blind alley? At some level, we cannot indict metaphysics without indicting the meaning of our existence.

While such a concern may seem to be merely motivating rhetoric for the critique, the consistency that this concern has with the type of solution Kant ultimately comes to give reason to believe it is not merely rhetoric. What if our own nature is subject to misinterpretation and our misunderstanding only shows itself later through contradictions that seem unavoidable? A paradigm shift resulting from a reinterpretation of our nature will be required to put us on the correct course. Kant's critique is designed to show the contradictions that have emerged naturally for us and present an option that avoids these issues. Furthermore, it will present a new option for interpreting our nature: that we are not meant to pursue knowledge of these metaphysical questions, but our nature is pushing us towards beliefs.

One more general note on the interpretation of our nature. This theme, which is akin to the maxim to "know thyself", contains an existentialist dimension of Kant. Here there is no grounding logical principle and instead one encounters a decision about the meaning of our lives that can change everything for us. Kant rarely works out these problems on the page, but it is worth pointing it out even just to show the limits of the scope of critical philosophy as well as to enable comparative philosophy that can deepen our insight (e.g., comparisons to Kierkegaard, Nietzsche, etc).

Terminology

speculative cognition of reason (spekulativen vernunfterkenntnis)

B xii-xiv, ¶ 7-8

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[¶7] It took natural science much longer to find the highway of science; for it is only about one and a half centuries since the suggestion of the ingenious Francis Bacon partly occasioned this discovery and partly further stimulated it, since one was already on its tracks - which discovery, therefore, can just as much be explained by a sudden revolution in the way of thinking. Here I will consider natural science only insofar as it is grounded on empirical principles.

[¶8] When Galileo rolled balls of a weight chosen by himself down an inclined plane, or when Torricelli made the air bear a weight that he had previously thought to be equal to that of a known column of water, or when in a later time Stahl changed metals into calx and then changed the latter back into metal by first removing something and then putting it back again,* a light dawned on all those who study nature. They comprehended that reason has insight only into what it itself produces according to its own design; that it must take the lead with principles for its judgments according to constant laws and compel nature to answer its questions, rather than letting nature guide its movements by keeping reason, as it were, in leading-strings; for otherwise accidental observations, made according to no previously designed plan, can never connect up into a necessary law, which is yet what reason seeks and requires. Reason, in order to be taught by nature, must approach nature with its principles in one hand, according to which alone the agreement among appearances can count as laws, and, in the other hand, the experiments thought out in accordance with these principles - yet in order to be instructed by nature not like a pupil, who has recited to him whatever the teacher wants to say, but like an appointed judge who compels witnesses to answer the questions he puts to them. Thus even physics owes the advantageous revolution in its way of thinking to the inspiration that what reason would not be able to know of itself and has to learn from nature, it has to seek in the latter (though not merely ascribe to it) in accordance with what reason itself puts into nature. This is how natural science was first brought to the secure course of a science after groping about for so many centuries.

* Here I am not following exactly the thread of the history of the experimental method, whose first beginnings are also not precisely known.

Summary

Natural science has also attained security as mathematics has by recognizing that progress can only be made by guiding or structuring observations in advance (a priori) rather than on the basis of chance perceptions. We can see this when looking at experiments within the study of nature.

Commentary

Natural science deploys concepts that are drawn from experience and must apply in experience while mathematics, on the other hand, constructs its concepts. Just as mathematics became secure by actively introducing its own principles, natural science's security is tied to our active participation in the process of knowing. This includes developing theories, hypotheses, and controls so we have a reference point that we understand in advance; these are so many elements of experiment and observation that we have not had to revise even as our knowledge of nature advances.

If we consider what Kant emphasizes in his examples, it is the active role that the scientists have in their experiments. We can certainly see that things fall, and even that things fall at different speeds (a feather falls relatively slowly), but if we do not experiment we cannot isolate what features are the cause of the difference in the rate objects fall. Kant means to emphasize this by mentioning Galileo's fabled experiment where there was an attempt to observe the movement of objects of different weights in a manner in which the weight of the objects was more likely to be the only contributing factor. If one didn't try to control the experiment, then any result we observe may not give us information since we don't know what things are kept the same or allowed to be different, and so we do not know what the different outcomes are tied to.

As the topic of natural science and nature arises here, it would be prudent to explain Kant's understanding of nature. Many modern readers, but not all, may have a tendency to think of natural science as studying nature as it is in itself. If this is the approach to the study of nature, then Kant would designate it as dogmatic metaphysics rather than natural science. Of course, modern physics is not dogmatic metaphysics and so cannot be a study of things in themselves. Just as in Kant, the study of nature is a study of objects so far as they can appear, directly or indirectly, so as to be observed. Our observations are possible because we are active - we run experiments, we have hypotheses and theories that anticipate what will happen, etc. An important takeaway from this is that in our model of natural science we design and apply the law to nature. It will be important to keep this aspect of the laws of nature in mind: we are the lawgivers in the sense that we design and apply the laws, and continue to adjust them as we are further instructed by experience.

Natural laws - or laws of any kind - are not observable per se: we can only observe singular or particular objects, while laws are universal and necessary. Instead, we observe the conformity of objects in nature to a law that we have given, or, in more general terms, we observe that our concept can be applied in experience. For example, we do not observe universal gravitation for the same reason we can't observe anything universal. However, we observe that objects conform to this law (our concept finds application in experience).

Kant will argue that any order of nature, and so all possible law for nature, is determined through the manner in which we construct experience. This is a point that will need to be developed further in the future, but which we should bear in mind.

The study of nature must not be confused with the study of things in themselves. Instead, the association we make with nature in Kant should be with the sum total of appearances. Appearances have a dimension that is provided from our sensible faculty, which is receptive, and which provides a relation we can think back to the object per se.

Questions

If we give the law to nature, how is it that natural science is discovered rather than invented?

The brief response to this is that even though we apply the law to nature, it isn't any law that will apply, ultimately we are looking around for something we can use to generally describe the behavior of nature which is considered something other than us. We may feel inclined to believe we are slowly approximating to the way nature is in itself, but we never leave the realm of appearances and the way we structure them so we never have an opportunity to see how close or far we have approached to knowing things themselves. It is always possible that nature, in itself, is completely lawless and that we provide any semblance of order we find in nature. However, whether nature in itself is ordered or not is a speculative proposition that exceeds our capacity.

Terminology

empirical principles (empirische Prinzipien)

B x-xii, ¶ 5-6

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[¶5] Mathematics and physics are the two theoretical cognitions of reason that are supposed to determine their objects a priori, the former entirely purely, the latter at least in part purely but also following the standards of sources of cognition other than reason.

[¶6] Mathematics has, from the earliest times to which the history of human reason reaches, in that admirable people the Greeks, traveled the secure path of a science. Yet it must not be thought that it was as easy for it as for logic - in which reason has to do only with itself - to find that royal path, or rather itself to open it up; rather, I believe that mathematics was left groping about for a long time (chiefly among the Egyptians), and that its transformation is to be ascribed to a revolution, brought about by the happy inspiration of a single man in an attempt from which the road to be taken onward could no longer be missed, and the secure course of a science was entered on and prescribed for all time and to an infinite extent. The history of this revolution in the way of thinking - which was far more important than the discovery of the way around the famous Cape - and of the lucky one who brought it about, has not been preserved for us. But the legend handed down to us by Diogenes Laertius - who names the reputed inventor of the smallest elements of geometrical demonstrations, even of those that, according to common judgment, stand in no need of proof - proves that the memory of the alteration wrought by the discovery of this new path in its earliest footsteps must have seemed exceedingly important to mathematicians, and was thereby rendered unforgettable. A new light broke upon the first person who demonstrated the isosceles triangle (whether he was called "Thales" or had some other name). For he found that what he had to do was not to trace what he saw in this figure, or even trace its mere concept, and read off, as it were, from the properties of the figure; but rather that he had to produce the latter from what he himself thought into the object and presented (through construction) according to a priori concepts, and that in order to know something securely a priori he had to ascribe to the thing nothing except what followed necessarily from what he himself had put into it in accordance with its concept.

Summary

Mathematics and physics are also examples of disciplines attained security as sciences. Mathematics attained security by attending to the properties that we introduce, a priori, into our own judgments and from which the a priori necessity of the mathematical cognitions extends.

Commentary

After lengthy praise of the significance of the revolution in mathematics, Kant discusses a crucial aspect of math's success: the necessity of our judgments in mathematics rest on what we have contributed to the construction of their concepts.

Kant recognized that advances in sciences followed the recognition that we must guide our study rather than depend on experience to lead us on its own. When mathematics became a secure science it made use of general solutions to problems a proiri and left behind imprecise measures such as using precalculated charts to approximate answers. Perhaps most memorable are the advances of ancient Greek geometry which must be limited in dealing with shapes so far as we experience them, but can judge universally and with necessity with concepts of figures such as circles and triangles constructed by us in advance.

As an example we can consider the first proposition in Euclid's Elements: to construct an equilateral triangle upon a given line. The proof follows by constructing more figures whose properties we understand in advance, namely circles. Two circles are constructed both with their center upon different ends of the line and with radii equal to the length of the entire given line. Because we know that the radii of the circles are the same as each other, and the same as the line they are constructed upon, we can find the place these circles intersect and connect each end of the line to this intersection point. The purpose of all this is to point out how the geometer does not sit idle, but actively contributes not only constructions to the proof, but the concepts that allow for the necessity resulting judgment. (An example such as this can also be used to point out how the judgments in mathematics are synthetic.)

Questions

Does Kant take mathematics to have been invented?

Concerning the debate of whether mathematics is invented or discovered, Kant appears to be on the side of invention; Kant even calls all knowledge from the construction of concepts mathematical knowledge. A brief argument for this can be drawn from how mathematics could attain a secure status. Mathematics judges with necessity and so these judgments can never be a posterioiri, but always a priori. The concepts of mathematics may have been inspired by reflection on experience in many cases, but many of these could not have been drawn from experience. The pure concepts of mathematics also require no exhibition in experience for their validity, for example, I do not need to discover a perfect circle or equilateral triangle for my concept of a circle to find acceptance - even in the application to calculations in experience. These concepts would then either be innate or constructed. If they are innate we would not learn mathematics as much as become conscious of it in ourselves, but this is not so, therefore these concepts are constructed by us.

To the argument that because everyone agrees on mathematics it must be discovered rather than invented, Kant's response would seem to be this: an alternative for the consistency in the judgments of mathematicians is also through the commonality of the form of intuition through which the concepts are constructed. The agreement between the mathematicians doesn't imply some inherent a priori insight into the order of the universe, but ultimately a form of social agreement around having a common sense.

Terminology

theoretical cognitions of Reason (theoretische Erkentnisse der Vernunft), pure (rein)