5.1The claim
In the last two paragraphs of Part I of Section IV, Hume separates two uses of reasoning in natural philosophy. One is the discovery of laws, which belongs wholly to experience. The other is the application of a law already found to a particular case, where mathematics gives exact dimensions. The separation matters for any decision that sets a size by calculation. The calculation may be exact, but it rests on a law, or an estimate, that only experience supplied, and no exactness in the calculation improves that foundation. This section reads paragraphs 26 and 27, with a part of paragraph 31, and then brings in the record Split entry or averaging. What it adds to the debate of Section 2 is a way of reading Kelly's arithmetic beside Hume: the sizing rule is a piece of what Hume calls mixed mathematics, exact in application and dependent on experience for its premise. The record shows a decision that refuses to let an improved average stand in for that premise.
5.2Staving off ignorance
Paragraph 26 states what the best natural philosophy can do and where it stops:
It is confessed, that the utmost effort of human reason is to reduce the principles, productive of natural phenomena, to a greater simplicity, and to resolve the many particular effects into a few general causes, by means of reasonings from analogy, experience, and observation.
The sentence is modest about method and generous about its results. Reason can reduce many effects to a few causes. It does so by analogy, experience and observation, not by deduction from first principles. Then Hume marks the limit:
The most perfect philosophy of the natural kind only staves off our ignorance a little longer: as perhaps the most perfect philosophy of the moral or metaphysical kind serves only to discover larger portions of it.
The phrase "staves off our ignorance a little longer" is about time. It describes knowledge not as a possession but as a delay: the best account postpones the point at which we must confess we do not know, without removing it. For a decision under uncertainty this gives a way of describing what a model or a plan does. It does not eliminate the unknown. It pushes the unknown further out, to a point the model does not cover. A position sized by a rule is safe within the range the rule was built on and exposed beyond it.
5.3Mixed mathematics
Paragraph 27 is the one that bears most directly on the arithmetic of sizing. Hume asks whether geometry, added to natural philosophy, can remedy the defect he has described. His answer is no, and the reason he gives separates two tasks:
Every part of mixed mathematics proceeds upon the supposition that certain laws are established by nature in her operations; and abstract reasonings are employed, either to assist experience in the discovery of these laws, or to determine their influence in particular instances, where it depends upon any precise degree of distance and quantity.
The first task is to assist experience in discovering laws. The second is to determine their influence in a particular case, where the answer depends on exact quantities. Hume illustrates the second with the law of momentum and the machines that exploit it, and then states the division in its sharpest form:
Geometry assists us in the application of this law, by giving us the just dimensions of all the parts and figures which can enter into any species of machine; but still the discovery of the law itself is owing merely to experience, and all the abstract reasonings in the world could never lead us one step towards the knowledge of it.
The sentence fits the Kelly criterion closely, though Hume could not have had it in view. Kelly's rule, as the note for his paper gives it, stakes a fraction set by the edge. This article adds that the rule can only be applied once the edge is known, and that applying it is an application in Hume's sense: it gives the just dimension of the stake, exact to any desired precision. The edge itself is the law. In Kelly's setting it is supplied by a private source of information. In a market it must be estimated, and the estimate comes from experience: from past trades, past prices, past backtests. All the exactness of the sizing formula, in Hume's phrase, "could never lead us one step towards the knowledge of it". A fraction computed from a mistaken edge is exact and wrong.
5.4Eggs and the first instance
The last passage this section reads comes from Part II, paragraph 31. Hume has asked why, if inference from experience were a matter of reasoning, it should need many instances rather than one:
Now it seems evident that, if this conclusion were formed by reason, it would be as perfect at first, and upon one instance, as after ever so long a course of experience. But the case is far otherwise. Nothing so like as eggs; yet no one, on account of this appearing similarity, expects the same taste and relish in all of them.
The eggs make a point the companion article did not take up. Resemblance in appearance is not enough even for ordinary expectation. Eggs look alike, and nobody expects them all to taste alike. What makes an expectation firm is a long course of uniform experience, not the likeness of one instance to another. For a trader whose position has moved against the plan, the eggs pose a sharp question. The second entry at a lower price looks like the first entry. It is the same instrument, the same direction, perhaps the same reason. But the market that has moved against the first entry is, in a respect that matters, a different market from the one in which the first entry was planned. Its appearance is similar. Whether its powers are the same is what the loss has put in doubt.
5.5The record: adding to a losing position
Synthetic scene: a modern summary written for the platform; the trader, the position and the market are invented.
The record Split entry or averaging opens with an assumption that removes the plan from the decision:
Assumption: The price, quantity, and final risk of additional orders were not fixed in advance, and the position size is to be increased after a loss occurs.
Two facts are stated. The additional orders had no fixed price, quantity or final risk before the trade began; and the size is to be increased after a loss. The first fact means that no law, in Hume's sense, was set down in advance for the additional orders. The second means that the step is being proposed in response to an outcome, the loss, rather than as part of a design. The record's question asks what tells the two apart:
What distinguishes an additional entry that is a planned split buy from loss-averaging that seeks to recover a loss?
The question names two acts that look alike. A planned split buy and loss-averaging both add to a position after the price has moved. In appearance, to use Hume's word, they are similar. The question is what difference lies beneath the appearance. The options are:
Additional entry
Decide only after confirming the total-risk cap
Prohibit additional entry
The second option is the record's form of a staged decision. It does not prohibit the addition, and it does not allow it straight away. It puts a calculation in front of it. The example judgment says what the calculation is:
An additional entry without a pre-planned plan and an overall risk cap is not called a planned split. The loss on invalidation of the entire position is calculated ahead of improving the average price.
The first sentence answers the question by definition. What separates a planned split from loss-averaging is not the act but what preceded it: a plan made beforehand, and a cap on the whole risk. Without those, the addition is not called a planned split, whatever it looks like. The judgment locates the difference where Hume locates it in the case of the eggs, in something that the appearance of the act does not show.
The second sentence sets an order of calculation. The loss that would follow if the whole position proved wrong is to be calculated before the improvement in average price. This is a direction to apply exact arithmetic to the worst case before the favourable one. It is mixed mathematics in Hume's sense: an exact determination of a quantity in a particular instance. What it does not do, and cannot do, is say whether the position is wrong. That is the law, and only experience will show it.
The record's Scenario field lists the evidence the judgment requires:
Required Evidence: Split plan before entry · Total exposure and average price after the additional order · Total loss if the position is finally invalidated
The first item is a plan that must exist before entry, which is to say before the experience that the loss supplies. The second and third are calculations. The record asks for both the average price, which the averaging trader wants to improve, and the total loss on invalidation, which the averaging trader has reason not to look at.
5.6The size of a step taken after a loss
The record connects to Kelly's argument at the point where an addition raises the stake. The note on Kelly's paper says that staking more than the fraction set by the edge lowers long-run growth and raises the risk of ruin. An addition to a losing position, made without a plan, raises the stake at a moment when the evidence about the edge has just turned against the trader. If the original stake was sized to the estimated edge, the loss is experience that bears on that estimate, and on a Humean reading it bears in the wrong direction for an increase. The record does not use Kelly's terms. It does not mention an edge or a fraction. But its demand that the total loss on invalidation be computed first, and that a cap on total risk be confirmed before any addition, has the effect of keeping the stake from rising above a bound set before the loss.
The step is irreversible in a particular way. Both of the parent study's coders read the harm of an addition as costly to undo: an added position can be closed, but only at the price then available, and closing a deepened loss realises it. The addition does not destroy an option in the clean sense of the option-value literature; it makes the exit dearer. Here, unlike the first case, the market supplies information whether the step is taken or not, through the further movement of the price. That is the condition under which the option-value argument fits best, and the record's third option, to prohibit the addition, is the form waiting takes here.
The record does not choose prohibition. Its example judgment sets the conditions under which an addition could count as planned. That leaves the addition open, on terms. The difference between the two cases is worth stating plainly. In the first case, the information that matters can come only from operation, and the record divides the step. In the second, information about the price will come in any case, and the record neither forbids the step nor waits; it puts a calculation and a plan in front of it. In neither case does it simply hold the step back.
5.7What the case adds
The case adds to the debate of Section 2 a reading of the sizing arithmetic through Hume's paragraph 27: exact calculation applies a law but cannot discover it, and in a market the law behind the Kelly fraction, the edge, is an estimate from experience. The record Split entry or averaging puts that division into practice. It treats the improved average price, an exact and attractive calculation, as subordinate to the total loss on invalidation, and it treats the difference between a planned split and loss-averaging as a difference in what came before the act, not in the act's appearance. Where the first case placed the record near Collingridge, this one places it between the two sides: the market supplies information whether or not the trader acts, as the option-value argument requires, and the record's answer is still not delay but a calculation and a plan set in front of the step. The record names no decider, no deadline and no number for the cap; the cap is to be confirmed, not stated.