Convertible Arbitrage. Thought of the Day 108.0

A convertible bond can be thought of as a fixed income security that has an embedded equity call option. The convertible investor has the right, but not the obligation, to convert (exchange) the bond into a predetermined number of common shares. The investor will presumably convert sometime at or before the maturity of the bond if the value of the common shares exceeds the cash redemption value of the bond. The convertible therefore has both debt and equity characteristics and, as a result, provides an asymmetrical risk and return profile. Until the investor converts the bond into common shares of the issuer, the issuer is obligated to pay a fixed coupon to the investor and repay the bond at maturity if conversion never occurs. A convertible’s price is sensitive to, among other things, changes in market interest rates, credit risk of the issuer, and the issuer’s common share price and share price volatility.

Untitled

Analysis of convertible bond prices factors in three different sources of value: investment value, conversion value, and option value. The investment value is the theoretical value at which the bond would trade if it were not convertible. This represents the security’s floor value, or minimum price at which it should trade as a nonconvertible bond. The conversion value represents the value of the common stock into which the bond can be converted. If, for example, these shares are trading at $30 and the bond can convert into 100 shares, the conversion value is $3,000. The investment value and conversion value can be considered, at maturity, the low and high price boundaries for the convertible bond. The option value represents the theoretical value of having the right, but not the obligation, to convert the bond into common shares. Until maturity, a convertible trades at a price between the investment value and the option value.

A Black-Scholes option pricing model, in combination with a bond valuation model, can be used to price a convertible security. However, a binomial option model, with some adjustments, is the best method for determining the value of a convertible security. Convertible arbitrage is a market-neutral investment strategy that involves the simultaneous purchase of convertible securities and the short sale of common shares (selling borrowed stock) that underlie the convertible. An investor attempts to exploit inefficiencies in the pricing of the convertible in relation to the security’s embedded call option on the convertible issuer’s common stock. In addition, there are cash flows associated with the arbitrage position that combine with the security’s inefficient pricing to create favorable returns to an investor who is able to properly manage a hedge position through a dynamic hedging process. The hedge involves selling short a percentage of the shares that the convertible can convert into based on the change in the convertible’s price with respect to the change in the underlying common stock price (delta) and the change in delta with respect to the change in the underlying common stock (gamma). The short position must be adjusted frequently in an attempt to neutralize the impact of changing common share prices during the life of the convertible security. This process of managing the short position in the issuer’s stock is called “delta hedging.”

If hedging is done properly, whenever the convertible issuer’s common share price decreases, the gain from the short stock position should exceed the loss from the convertible holding. Equally, whenever the issuer’s common share price increases, the gain from the convertible holding should exceed the loss from the short stock position. In addition to the returns produced by delta hedging, the investor will receive returns from the convertible’s coupon payment and interest income associated with the short stock sale. However, this cash flow is reduced by paying a cash amount to stock lenders equal to the dividend the lenders would have received if the stock were not loaned to the convertible investor, and further reduced by stock borrow costs paid to a prime broker. In addition, if the investor leverages the investment by borrowing cash from a prime broker, there will be interest expense on the loan. Finally, if an investor chooses to hedge credit risk of the issuer, or interest rate risk, there will be additional costs associated with credit default swaps and a short Treasury position. This strategy attempts to create returns that exceed the returns that would be available from purchasing a nonconverting bond with the same maturity issued by the same issuer, without being exposed to common share price risk. Most convertible arbitrageurs attempt to achieve double-digit annual returns from convertible arbitrage.

Advertisements

(Il)liquid Hedge Lock-Ups. Thought of the Day 107.0

here-are-the-hedge-funds-that-are-dominating-in-2016

Hedge funds have historically limited their participation in illiquid investments, preferring to match their investment horizon to the typical one-year lock-up periods that their investors agree to. However, many hedge funds have increasingly invested in illiquid assets in an effort to augment returns. For example, they have invested in private investments in public equity (PIPEs), acquiring large minority holdings in public companies. Their purchases of CDOs and CLOs (collateralized loan obligations) are also somewhat illiquid, since these fixed income securities are difficult to price and there is a limited secondary market during times of crisis. In addition, hedge funds have participated in loans, and invested in physical assets. Sometimes, investments that were intended to be held for less than one year have become long-term, illiquid assets when the assets depreciated and hedge funds decided to continue holding the assets until values recovered, rather than selling at a loss. It is estimated that more than 20% of total assets under management by hedge funds are illiquid, hard-to-price assets. This makes hedge fund asset valuation difficult, and has created a mismatch between hedge fund assets and liabilities, giving rise to significant problems when investors attempt to withdraw their cash at the end of lock-up periods.

Hedge funds generally focus their investment strategies on financial assets that are liquid and able to be readily priced based on reported prices in the market for those assets or by reference to comparable assets that have a discernible price. Since most of these assets can be valued and sold over a short period of time to generate cash, hedge funds permit investors to invest in or withdraw money from the fund at regular intervals and managers receive performance fees based on quarterly mark-to-market valuations. However, in order to match up maturities of assets and liabilities for each investment strategy, most hedge funds have the ability to prevent invested capital from being withdrawn during certain periods of time. They achieve this though “lock-up” and “gate” provisions that are included in investment agreements with their investors.

A lock-up provision provides that during an initial investment period of, typically, one to two years, an investor is not allowed to withdraw any money from the fund. Generally, the lock-up period is a function of the investment strategy that is being pursued. Sometimes, lock-up periods are modified for specific investors through the use of a side letter agreement. However, this can become problematic because of the resulting different effective lock-up periods that apply to different investors who invest at the same time in the same fund. Also, this can trigger “most favored nations” provisions in other investor agreements.

A gate is a restriction that limits the amount of withdrawals during a quarterly or semi- annual redemption period after the lock-up period expires. Typically gates are percentages of a fund’s capital that can be withdrawn on a scheduled redemption date. A gate of 10 to 20% is common. A gate provision allows the hedge fund to increase exposure to illiquid assets without facing a liquidity crisis. In addition, it offers some protection to investors that do not attempt to withdraw funds because if withdrawals are too high, assets might have to be sold by the hedge fund at disadvantageous prices, causing a potential reduction in investment returns for remaining investors. During 2008 and 2009, as many hedge fund investors attempted to withdraw money based on poor returns and concerns about the financial crisis, there was considerable frustration and some litigation directed at hedge fund gate provisions.

Hedge funds sometimes use a “side pocket” account to house comparatively illiquid or hard-to-value assets. Once an asset is designated for inclusion in a side pocket, new investors don’t participate in the returns from this asset. When existing investors withdraw money from the hedge fund, they remain as investors in the side pocket asset until it either is sold or becomes liquid through a monetization event such as an IPO. Management fees are typically charged on side pocket assets based on their cost, rather than a mark-to-market value of the asset. Incentive fees are charged based on realized proceeds when the asset is sold. Usually, there is no requirement to force the sale of side pocket investments by a specific date. Sometimes, investors accuse hedge funds of putting distressed assets that were intended to be sold during a one-year horizon into a side pocket account to avoid dragging down the returns of the overall fund. Investors are concerned about unexpected illiquidity arising from a side pocket and the potential for even greater losses if a distressed asset that has been placed there continues to decline in value. Fund managers sometimes use even more drastic options to limit withdrawals, such as suspending all redemption rights (but only in the most dire circumstances).

Hans-Hermann Hoppe, Libertarianism and the “Alt-Right” (PFS 2017)

A new victimology has been proclaimed and promoted. Women — and in particular single mothers — blacks, browns, Latinos, homosexuals, lesbians, bi, and transsexuals have been awarded victim status, and accorded legal privileges through nondiscrimination or affirmative action decrees as well. Most recently such privileges have been expanded also to foreign national immigrants, whether legal or illegal, insofar as they fall into one of the just mentioned categories, or are members of non-Christian religions such as Islam for instance.

Hoppe does not identify as alt-right, but runs in the same circles as prominent white nationalists. His popularity among fringe anarcho-capitalists – or ancaps – has resulted in a plethora of memes, sometimes depicting Hoppe as Pepe the Frog, and often bearing the slogan “Hippity Hoppity, Get Off My Property.” One of Hoppe’s proposals – that truly libertarian societies be able to “physically remove” Communists and other undesirables from their ranks – has become a meme on the far-right thanks to the “Crying Nazi” himself, Christopher Cantwell. His online store stocks “I ♥ Physical Removal” stickers, along with a “Right-Wing Death Squad” hat, and Radical Agenda shirts depicting a person being thrown from a helicopter – in honor of Augusto Pinochet….

Hoppe told his audience that “many of the leading lights associated with the alt-right have appeared here at our meetings in the course of the years,” including Paul Gottfried, Peter Brimelow, Richard Lynn, Jared Taylor, John Derbyshire, Steve Sailer, and Richard Spencer. And he boasted that these associations have “earned” him “several honorable mentions” by the SPLC, which he called “America’s most famous smear and defamation league.”

According to Hoppe, “many libertarians” are “plain ignorant of human psychology and sociology” and “devoid of any common sense.” He said this explains their tendency to “blindly accept, against all empirical evidence, an egalitarian, blank slate view of human nature that all people and all societies and all cultures are essentially equal and interchangeable.”

The alt-right, on the other hand, does not labor under such delusions. He described the alt-right as “united” in its “identification and diagnosis of [the West’s] social pathologies.” The alt-right is “against, and indeed it hates with a passion, the elites in control of the State, the mainstream media, and academia” because they promote “egalitarianism, affirmative action or nondiscrimination laws, multiculturalism, and free mass immigration as a means to bring about this multiculturalism.”

 

Modal Structuralism. Thought of the Day 106.0

1c937ce49f02f66e2f61c08b75ba2709--labs-joseph

Structuralism holds that mathematics is ultimately about the shared structures that may be instantiated by particular systems of objects. Eliminative structuralists, such as Geoffrey Hellman (Mathematics Without Numbers Towards a Modal-Structural Interpretation), try to develop this insight in a way that does not assume the existence of abstract structures over and above any instances. But since not all mathematical theories have concrete instances, this brings a modal element to this kind of structuralist view: mathematical theories are viewed as being concerned with what would be the case in any system of objects satisfying their axioms. In Hellman’s version of the view, this leads to a reinterpretation of ordinary mathematical utterances made within the context of a theory. A mathematical utterance of the sentence S, made against the context of a system of axioms expressed as a conjunction AX, becomes interpreted as the claim that the axioms are logically consistent and that they logically imply S (so that, were we to find an interpretation of those axioms, S would be true in that interpretation). Formally, an utterance of the sentence S becomes interpreted as the claim:

◊ AX & □ (AX ⊃ S)

Here, in order to preserve standard mathematics (and to avoid infinitary conjunctions of axioms), AX is usually a conjunction of second-order axioms for a theory. The operators “◊” and “□” are modal operators on sentences, interpreted as “it is logically consistent that”, and “it is logically necessary that”, respectively.

This view clearly shares aspects of the core of algebraic approaches to mathematics. According to modal structuralism what makes a mathematical theory good is that it is logically consistent. Pure mathematical activity becomes inquiry into the consistency of axioms, and into the consequences of axioms that are taken to be consistent. As a result, we need not view a theory as applying to any particular objects, so certainly not to one particular system of objects. Since mathematical utterances so construed do not refer to any objects, we do not get into difficulties with deciding on the unique referent for apparent singular terms in mathematics. The number 2 in mathematical contexts refers to no object, though if there were a system of objects satisfying the second-order Peano axioms, whatever mathematical theorems we have about the number 2 would apply to whatever the interpretation of 2 is in that system. And since our mathematical utterances are made true by modal facts, about what does and does not follow from consistent axioms, we no longer need to answer Benacerraf’s question of how we can have knowledge of a realm of abstract objects, but must instead consider how we know these (hopefully more accessible) facts about consistency and logical consequence.

Stewart Shapiro’s (Philosophy of Mathematics Structure and Ontology) non-eliminative version of structuralism, by contrast, accepts the existence of structures over and above systems of objects instantiating those structures. Specifically, according to Shapiro’s ante rem view, every logically consistent theory correctly describes a structure. Shapiro uses the terminology “coherent” rather than “logically consistent” in making this claim, as he reserves the term “consistent” for deductively consistent, a notion which, in the case of second-order theories, falls short of coherence (i.e., logical consistency), and wishes also to separate coherence from the model-theoretic notion of satisfiability, which, though plausibly coextensive with the notion of coherence, could not be used in his theory of structure existence on pain of circularity. Like Hellman, Shapiro thinks that many of our most interesting mathematical structures are described by second-order theories (first-order axiomatizations of sufficiently complex theories fail to pin down a unique structure up to isomorphism). Mathematical theories are then interpreted as bodies of truths about structures, which may be instantiated in many different systems of objects. Mathematical singular terms refer to the positions or offices in these structures, positions which may be occupied in instantiations of the structures by many different officeholders.

While this account provides a standard (referential) semantics for mathematical claims, the kinds of objects (offices, rather than officeholders) that mathematical singular terms are held to refer to are quite different from ordinary objects. Indeed, it is usually simply a category mistake to ask of the various possible officeholders that could fill the number 2 position in the natural number structure whether this or that officeholder is the number 2 (i.e., the office). Independent of any particular instantiation of a structure, the referent of the number 2 is the number 2 office or position. And this office/position is completely characterized by the axioms of the theory in question: if the axioms provide no answer to a question about the number 2 office, then within the context of the pure mathematical theory, this question simply has no answer.

Elements of the algebraic approach can be seen here in the emphasis on logical consistency as the criterion for the existence of a structure, and on the identification of the truths about the positions in a structure as being exhausted by what does and does not follow from a theory’s axioms. As such, this version of structuralism can also respond to Benacerraf’s problems. The question of which instantiation of a theoretical structure one is referring to when one utters a sentence in the context of a mathematical theory is dismissed as a category mistake. And, so long as the basic principle of structure-existence, according to which every logically consistent axiomatic theory truly describes a structure, is correct, we can explain our knowledge of mathematical truths simply by appeal to our knowledge of consistency.

Price-Earnings Ratio. Note Quote.

The price-earnings ratio (P/E) is arguably the most popular price multiple. There are numerous definitions and variations of the price-earnings ratio. In its simplest form, the price-earnings ratio relates current share price to earnings per share.

Untitled

The forward (or estimated) price-earnings ratio is based on the current stock price and the estimated earnings for future full scal years. Depending on how far out analysts are forecasting annual earnings (typically, for the current year and the next two fiscal years), a company can have multiple forward price-earnings ratios. The forward P/E will change as earnings estimates are revised when new information is released and quarterly earnings become available. Also, forward price-earnings ratios are calculated using estimated earnings based on the current fundamentals. A company’s fundamentals could change drastically over a short period of time and estimates may lag the changes as analysts digest the new facts and revise their outlooks.

The average price-earnings ratio attempts to smooth out the price-earnings ratio by reducing daily variation caused by stock price movements that may be the result of general volatility in the stock market. Different sources may calculate this figure differently. Average P/E is defined as the average of the high and low price-earnings ratios for a given year. The high P/E is calculated by dividing the high stock price for the year by the annual earnings per share fully diluted from continuing operations. The low P/E for the year is calculated using the low stock price for the year.

The relative price-earnings ratio helps to compare a company’s price-earnings ratio to the price-earnings ratio of the overall market, both currently and historically. Relative P/E is calculated by dividing the firm’s price-earnings ratio by the market’s price-earnings ratio.

The price-earnings ratio is used to gauge market expectation of future performance. Even when using historical earnings, the current price of a stock is a compilation of the market’s belief in future prospects. Broadly, a high price-earnings ratio means the market believes that that the company has strong future growth prospects. A low price-earnings ratio generally means the market has low earnings growth expectations for the firm or there is high risk or uncertainty of the firm actually achieving growth. However, looking at a price-earnings ratio alone may not be too illuminating. It will always be more useful to compare the price-earnings ratios of one company to those of other companies in the same industry and to the market in general. Furthermore, tracking a stock’s price-earnings ratio over time is useful in determining how the current valuation compares to historical trends.

Gordon growth model is a variant of the discounted cash flow model, is a method for valuing intrinsic value of a stock or business. Many researches on P/E ratios are based on this constant dividend growth model.

When investors purchase a stock, they expect two kinds of cash flows: dividend during holding shares and expected stock price at the end of shareholding. As the expected share price is decided by future dividend, then we can use the unlimited discount to value the current price of stocks.

A normal model for the intrinsic value of a stock:

V = D1/(1+R)1 + D2/(1+R)2 +…+ Dn/(1+R)n = ∑t=1 Dt/(1+R)t (n→∞) —– (1)

In (1)

V: intrinsic value of the stock;

Dt: dividend for the tth year

R: discount rate, namely required rate of return;

t: the year for dividend payment.

Assume the market is efficient, the share price should be equal to the intrinsic value of the stock, then equation (1) becomes:

P0 = D1/(1+R)1 + D2/(1+R)2 +…+ Dn/(1+R)n = ∑t=1 Dt/(1+R)t (n→∞) —– (2)

where P0: purchase price of the stock;

Dt: dividend for the tth year

R: discount rate, namely required rate of return;

t: the year for dividend payment.

Assume the dividend grows stably at the rate of g, we derive the constant dividend growth model.

That is Gordon constant dividend growth model:

P0 = D1/(1+R)1 + D2/(1+R)2 +…+ Dn/(1+R)n = D0(1+g)/(1+R)1 + D0(1+g)2/(1+R)2 +….+ D0(1+g)n/(1+R)n = ∑t=1 D0(1+g)t/(1+R)t —– (3)

When g is a constant, and R>g at the same time, then equation (3) can be modified as the following:

P0 = D0(1+g)/(R-g) = D1/(R-g) —– (4)

where, P0: purchase price of the stock;

D0: dividend at the purchase time;

D1: dividend for the 1st year;

R: discount rate, namely required rate of return;

g: the growth rate of dividend.

We suppose that the return on dividend b is fixed, then equation (4) divided by E1 is:

P0/E1 = (D1/E1)/(R-g) = b/(R-g) —– (5)

where, P0: purchase price of the stock;

D1: dividend for the 1st year;

E1: earnings per share (EPS) of the 1st year after purchase;

b: return on dividend;

R: discount rate, namely required rate of return;

g: the growth rate of dividend.

Therefrom we derive the P/E ratio theoretical computation model, from which appear factors deciding P/E directly, namely return on dividend, required rate of return and the growth rate of dividend. The P/E ratio is related positively to the return on dividend and required rate of return, and negatively to the growth rate of dividend.

Realistically speaking, most investors relate high P/E ratios to corporations with fast growth of future profits. However, the risk closely linked the speedy growth is also very important. They can counterbalance each other. For instance, when other elements are equal, the higher the risk of a stock, the lower is its P/E ratio, but high growth rate can counterbalance the high risk, thus lead to a high P/E ratio. P/E ratio reflects the rational investors’ expectation on the companies’ growth potential and risk in the future. The growth rate of dividend (g) and required rate of return (R) in the equation also response growth opportunity and risk factors.

Financial indices such as Dividend Payout Ratio, Liability-Assets (L/A) Ratio and indices that reflecting growth and profitability are employed in this paper as direct influence factors that have impact on companies’ P/E ratios.

Derived from (5), the dividend payout ratio has a direct positive effect on P/E ratio. When there is a high dividend payout ratio, the returns and stock value investors expected will also rise, which lead to a high P/E ratio. Conversely, the P/E ratio will be correspondingly lower.

Earnings per share (EPS) is another direct factor, while its impact on P/E ratio is negative. It reflects the relation between capital size and profit level of the company. When the profit level is the same, the larger the capital size, the lower the EPS will be, then the higher the P/E ratio will be. When the liability-assets ratio is high, which represents that the proportion of the equity capital is lower than debt capital, then the EPS will be high and finally the P/E ratio will led to be low. Therefore, the companies’ L/A ratio also negatively correlate to P/E ratio.

Some other financial indices including growth rate of EPS, ROE, growth rate of ROE, growth rate of net assets, growth rate of main business income and growth rate of main business profit should theoretically positively correlate to P/E ratios, because if the companies’ growth and profitability are both great, then investors’ expectation will be high, and then the stock prices and P/E ratios will be correspondingly high. Conversely, they will be low.

In the Gordon growth model, the growth of dividend is calculated based on the return on retained earnings reinvestment, r, therefore:

g = r (1-b) = retention ratio return on retained earnings.

As a result,

P0/E1 = b/(R-g) = b/(R-r(1-b)) —– (6)

Especially, when the expected return on retained earnings equal to the required rate of return (i.e. r = R) or when the retained earnings is zero (i.e. b=1),

There is:

P0/E1 = 1/R —– (7)

Obviously, in (7) the theoretical value of P/E ratio is the reciprocal of the required rate of return. According to the Capital Asset Pricing Model (CAPM), the average yields of the stock market should be equal to risk-free yield plus total risk premium. When there not exists any risk, then the required rate of return will equal to the market interest rate. Thus, the P/E ratio here turns into the reciprocal of the market interest rate.

As an important influence factor, the annual interest rate affect on both market average and companies’ individual P/E ratios. On the side of market average P/E ratio, when interest rate declines, funds will move to security markets, funds supply volume increasing will lead to the rise of share prices, and then rise in P/E ratios. In contrast, when interest rate rises, revulsion of capitals will reflow into banks, funds supply will be critical, share prices decline as well as P/E ratios. On the other side on the companies’ P/E ratio, the raise on interest rate will be albatross of companies, all other conditions remain, earnings will reduce, then equity will lessen, large deviation between operation performance and expected returns appears, can not support a high level of P/E ratio, so stock prices will decline. As a result, both market average and companies’ individual P/E ratios will be influenced by the annual interest rate.

It is also suitable to estimate the market average P/E ratio, and only when all the above assumptions are satisfied, that the practical P/E ratio amount to the theoretical value. However, different from the securities market, the interest rate is relatively rigid, especially to the strict control of interest rate countries; the interest rate adjustment is not so frequent, so that it is not synchronous with macroeconomic fundamentals. Reversely, the stock market reflects the macroeconomic fundamentals; high expectation of investors can raise up the stock prices, sequent the growth of the aggregate value of the whole market. Other market behaviors can also lead to changes of average P/E ratios. Therefore, it is impossible that the average P/E ratio is identical with the theoretical one. Variance exits inevitably, the key is to measure a rational range for this variance.

For the market average P/E ratio, P should be the aggregate value of listed stocks, and E is the total level of capital gains. To the maturity market, the reasonable average P/E ratio should be the reciprocal of the average yields of the market; usually the bank annual interest is used to represent the average yields of the market.

The return on retained earnings is an expected value in theory, while it is always hard to forecast, so the return on equity (ROE) is used to estimate the value.

(6) can then evolve as,

P0/E1 = b/(R-g) = b/(R-r(1-b)) = b/(R-ROE(1-b)) —– (8)

From (8) we can know, ROE is one of the influence factors to P/E ratio, which measures the value companies created for shareholders. It is positively correlated to the P/E ratio. The usefulness of any price-earnings ratio is limited to firms that have positive actual and expected earnings. Depending on the data source you use, companies with negative earnings will have a “null” value for a P/E while other sources will report a P/E of zero. In addition, earnings are subject to management assumptions and manipulation more than other income statement items such as sales, making it hard to get a true sense of value.

Quantifier – Ontological Commitment: The Case for an Agnostic. Note Quote.

1442843080570

What about the mathematical objects that, according to the platonist, exist independently of any description one may offer of them in terms of comprehension principles? Do these objects exist on the fictionalist view? Now, the fictionalist is not committed to the existence of such mathematical objects, although this doesn’t mean that the fictionalist is committed to the non-existence of these objects. The fictionalist is ultimately agnostic about the issue. Here is why.

There are two types of commitment: quantifier commitment and ontological commitment. We incur quantifier commitment to the objects that are in the range of our quantifiers. We incur ontological commitment when we are committed to the existence of certain objects. However, despite Quine’s view, quantifier commitment doesn’t entail ontological commitment. Fictional discourse (e.g. in literature) and mathematical discourse illustrate that. Suppose that there’s no way of making sense of our practice with fiction but to quantify over fictional objects. Still, people would strongly resist the claim that they are therefore committed to the existence of these objects. The same point applies to mathematical objects.

This move can also be made by invoking a distinction between partial quantifiers and the existence predicate. The idea here is to resist reading the existential quantifier as carrying any ontological commitment. Rather, the existential quantifier only indicates that the objects that fall under a concept (or have certain properties) are less than the whole domain of discourse. To indicate that the whole domain is invoked (e.g. that every object in the domain have a certain property), we use a universal quantifier. So, two different functions are clumped together in the traditional, Quinean reading of the existential quantifier: (i) to assert the existence of something, on the one hand, and (ii) to indicate that not the whole domain of quantification is considered, on the other. These functions are best kept apart. We should use a partial quantifier (that is, an existential quantifier free of ontological commitment) to convey that only some of the objects in the domain are referred to, and introduce an existence predicate in the language in order to express existence claims.

By distinguishing these two roles of the quantifier, we also gain expressive resources. Consider, for instance, the sentence:

(∗) Some fictional detectives don’t exist.

Can this expression be translated in the usual formalism of classical first-order logic with the Quinean interpretation of the existential quantifier? Prima facie, that doesn’t seem to be possible. The sentence would be contradictory! It would state that ∃ fictional detectives who don’t exist. The obvious consistent translation here would be: ¬∃x Fx, where F is the predicate is a fictional detective. But this states that fictional detectives don’t exist. Clearly, this is a different claim from the one expressed in (∗). By declaring that some fictional detectives don’t exist, (∗) is still compatible with the existence of some fictional detectives. The regimented sentence denies this possibility.

However, it’s perfectly straightforward to express (∗) using the resources of partial quantification and the existence predicate. Suppose that “∃” stands for the partial quantifier and “E” stands for the existence predicate. In this case, we have: ∃x (Fx ∧¬Ex), which expresses precisely what we need to state.

Now, under what conditions is the fictionalist entitled to conclude that certain objects exist? In order to avoid begging the question against the platonist, the fictionalist cannot insist that only objects that we can causally interact with exist. So, the fictionalist only offers sufficient conditions for us to be entitled to conclude that certain objects exist. Conditions such as the following seem to be uncontroversial. Suppose we have access to certain objects that is such that (i) it’s robust (e.g. we blink, we move away, and the objects are still there); (ii) the access to these objects can be refined (e.g. we can get closer for a better look); (iii) the access allows us to track the objects in space and time; and (iv) the access is such that if the objects weren’t there, we wouldn’t believe that they were. In this case, having this form of access to these objects gives us good grounds to claim that these objects exist. In fact, it’s in virtue of conditions of this sort that we believe that tables, chairs, and so many observable entities exist.

But recall that these are only sufficient, and not necessary, conditions. Thus, the resulting view turns out to be agnostic about the existence of the mathematical entities the platonist takes to exist – independently of any description. The fact that mathematical objects fail to satisfy some of these conditions doesn’t entail that these objects don’t exist. Perhaps these entities do exist after all; perhaps they don’t. What matters for the fictionalist is that it’s possible to make sense of significant features of mathematics without settling this issue.

Now what would happen if the agnostic fictionalist used the partial quantifier in the context of comprehension principles? Suppose that a vector space is introduced via suitable principles, and that we establish that there are vectors satisfying certain conditions. Would this entail that we are now committed to the existence of these vectors? It would if the vectors in question satisfied the existence predicate. Otherwise, the issue would remain open, given that the existence predicate only provides sufficient, but not necessary, conditions for us to believe that the vectors in question exist. As a result, the fictionalist would then remain agnostic about the existence of even the objects introduced via comprehension principles!

Fictionalism. Drunken Risibility.

mathematical-objects

Applied mathematics is often used as a source of support for platonism. How else but by becoming platonists can we make sense of the success of applied mathematics in science? As an answer to this question, the fictionalist empiricist will note that it’s not the case that applied mathematics always works. In several cases, it doesn’t work as initially intended, and it works only when accompanied by suitable empirical interpretations of the mathematical formalism. For example, when Dirac found negative energy solutions to the equation that now bears his name, he tried to devise physically meaningful interpretations of these solutions. His first inclination was to ignore these negative energy solutions as not being physically significant, and he took the solutions to be just an artifact of the mathematics – as is commonly done in similar cases in classical mechanics. Later, however, he identified a physically meaningful interpretation of these negative energy solutions in terms of “holes” in a sea of electrons. But the resulting interpretation was empirically inadequate, since it entailed that protons and electrons had the same mass. Given this difficulty, Dirac rejected that interpretation and formulated another. He interpreted the negative energy solutions in terms of a new particle that had the same mass as the electron but opposite charge. A couple of years after Dirac’s final interpretation was published Carl Anderson detected something that could be interpreted as the particle that Dirac posited. Asked as to whether Anderson was aware of Dirac’s papers, Anderson replied that he knew of the work, but he was so busy with his instruments that, as far as he was concerned, the discovery of the positron was entirely accidental.

The application of mathematics is ultimately a matter of using the vocabulary of mathematical theories to express relations among physical entities. Given that, for the fictionalist empiricist, the truth of the various theories involved – mathematical, physical, biological, and whatnot – is never asserted, no commitment to the existence of the entities that are posited by such theories is forthcoming. But if the theories in question – and, in particular, the mathematical theories – are not taken to be true, how can they be successfully applied? There is no mystery here. First, even in science, false theories can have true consequences. The situation here is analogous to what happens in fiction. Novels can, and often do, provide insightful, illuminating descriptions of phenomena of various kinds – for example, psychological or historical events – that help us understand the events in question in new, unexpected ways, despite the fact that the novels in question are not true. Second, given that mathematical entities are not subject to spatial-temporal constraints, it’s not surprising that they have no active role in applied contexts. Mathematical theories need only provide a framework that, suitably interpreted, can be used to describe the behavior of various types of phenomena – whether the latter are physical, chemical, biological, or whatnot. Having such a descriptive function is clearly compatible with the (interpreted) mathematical framework not being true, as Dirac’s case illustrates so powerfully. After all, as was just noted, one of the interpretations of the mathematical formalism was empirically inadequate.

On the fictionalist empiricist account, mathematical discourse is clearly taken on a par with scientific discourse. There is no change in the semantics. Mathematical and scientific statements are treated in exactly the same way. Both sorts of statements are truth-apt, and are taken as describing (correctly or not) the objects and relations they are about. The only shift here is on the aim of the research. After all, on the fictionalist empiricist proposal, the goal is not truth, but something weaker: empirical adequacy – or truth only with respect to the observable phenomena. However, once again, this goal matters to both science and (applied) mathematics, and the semantic uniformity between the two fields is still preserved. According to the fictionalist empiricist, mathematical discourse is also taken literally. If a mathematical theory states that “There are differentiable functions such that…”, the theory is not going to be reformulated in any way to avoid reference to these functions. The truth of the theory, however, is never asserted. There’s no need for that, given that only the empirical adequacy of the overall theoretical package is required.