Recipes

Detecting a recipe pattern requires an observation of the following kind:

Once certain ingredients are available, a specified series of actions leads to some result.

By formulating a recipe, we can communicate and conserve experience which can be used by anyone who is able to apply the recipe by matching actual object constellations to the ingredients.

Recipe formulation

A recipe can be split into three parts: (1) the setting describes the required ingredients together with properties which ensure that the intended actions are successfully executable. (2) The development consists of a sequence of actions that are carried out with the ingredients. (3) The result singles out certain inevitable consequences of the actions.

In a typical baking recipe, obvious ingredients are certain amounts of flour, butter eggs and sugar. But there are also ingredients that are usually not mentioned like
a blender, a bowl a cake pan and an oven. As a result of the described actions
the cake is typically highlighted while the inevitably dirty bowl and the increased
electricity bill are usually not mentioned.

Recipe application

In order to apply a recipe, a constellation of objects is required which fits to the recipe’s setting. If all requirements are met, the described sequence of actions
can be executed and the indicated result is obtained.

Detecting that a specific object constellation fits to a setting is an action which could be called pattern matching. It results in a one-to-one identification between ingredients and objects so that all ingredient conditions are satisfied by the associated objects in a corresponding sense.

If our brain were unable to carry out this matching process we would be incapable of applying recipes. Conversely, finding recipes requires pattern recognition. In this case, the reproducibility of a certain result using particular actions in a given setting has to be recognized as a pattern.

The collection of all object constellations which match a setting is called
the recipe’s range of applicability. While this range is huge, for example, in the case of crafting instructions for straw stars due to the almost unlimited supply of straw, sewing thread and scissors, it can also be small like in recipes about proper interactions between political leaders of different countries.

Recipe results

In the case of baking recipes, we may be eager to apply them because the results are delicious cakes or cookies. In other cases, the results are undesirable (like being hit
by lightning on a mountain top or in a lake) and we try to avoid being part of the ingredients. In both cases, however, it is useful to know the recipes and to understand their application process.

We also see that the result of recipes may be of different nature· Some recipes combine the ingredients to form new objects while other recipes just ensure a certain state of an ingredient (being shiny in the case of polishing recipes).

Further recipes of the second kind are rules: “When you want to cross a street, pay attention to the traffic first!” Here, the unspoken result is a preferable state of being unharmed. Similarly, in rules of politieness, the ingredients are human beings in various scenarios where the specified actions ensure the state “pleased” or “comfortable” or at least “not offended” of the target person(s).

The special role of settings

Among the three parts of a recipe, the setting is the most characteristic one. In fact, a text which simply states the results of some actions applied to specific objects would be called a report or a protocol but not a recipe. Only if the actions happen in an ingredient setting, their potential future use is ensured. In view of this, it is fair to say that the recipe concept is inextricably coupled to its setting part.

But a setting part alone does not constitute a recipe. It merely singles out object constellations with a certain potential. Only the development and result parts clarify how the potential can be used by shifting the focus to certain inherent features.

If the same setting appears in several recipes, its admissible objects are interesting in various aspects and it may be worthwhile to group them under a general term. In this way, settings can be used as generators of terms which then abbreviate the formulation of recipes and alleviate the sharing of experience.

Fiction

Since every recipe begins with the assumption that certain ingredients are present, the entire text is automatically fictitious. This is further exacerbated if the assumed properties of the ingredients can only be fulfilled with great difficulty or not at all by a real object constellation. But even if a recipe’s range of applicability is empty, the development and the result achieved can still be very reasonable, assuming that suitable ingredients were available.

At first glance it seems that there is no direct usage of recipes with an empty range of applicability because we can never find objects that meet the requirements exactly. But if we think about it more thorougly, we discover that we are actually using such recipes all the time. A baking recipe which requires one egg is a simple example. Although eggs come in different sizes and with different yolk to white ratios we simply grab an egg from the fridge without hesitation because we know from experience, that the recipe is robust – the result doesn’t change much if the ingredients vary a little.

In fact, as words in natural languages are rarely defined precisely, it is impossible to satisfy setting conditions exactly. Similarly, the actions listed in the development are typically lacking a precise definition so that we cannot follow them exactly either.

Like any text, a recipe is merely an idealized pattern that can be reproduced physically or mentally if the symbols in the text are suitably interpreted. In this sense, even purely fictional recipes (with laser swords and the like) can still convey experiences – ultimately, it all depends on how we interpret the ingredients and actions.

Everyone knows recipes

Recipes as embodiments of human experience are constantly around us in the form of instructions, rules and advice. Dealing with recipes requires a few basic actions (highlighted in red) which refer to a few characteristic notions (highlighted in blue):

• Detecing a recipe pattern requires a sense for the reproducibility of a result using particular actions in a given setting.
• Accordingly, the formulation of a recipe may be split into three parts: setting, development and result.
• In a setting, ingredients are introduced and requirements are listed.
• Object constellations which can be matched to a setting are called admissible.
• The range of applicability summarizes all admissible constellations of a setting.
• Recipes can be applied to admissible constellations by carrying out the actions listed in the development.
• The result of a recipe may be an object or a state.
• Rules can be viewed as recipes whose results are states.

It is in the nature of things that children have less world experience than adults, so they are particularly often confronted with all kinds of recipes. In particular, children are well aware of this concept once they enter school and math education should use this awareness as recipes are at the very core of mathematics.

Mathematical recipes

Mathematical recipes are used to encode experience with mathematical objects. As with other recipes, their formulation can be split into setting, development and result. Moreover, by restricting the syntactic form of recipes, the role of each part can be described very accurately.

In fact, these language constructs are the most notable differences between mathematical and non-mathematical recipes.

Variables

As an illustration, we consider the first lines of some origami instruction:

You need a quadratic sheet of paper. Begin with folding the sheet in half…

In natural language, the article “a” introduces the ingredient “quadratic sheet of paper” indefinitely which indicates that any square sheet could be used in this case. After its introduction, it is then possible to refer to it with a definite article which can be seen in the development, where “the sheet” is used as reference. In other words, natural language employs noun phrases with indefinite articles to stage the ingredients (possibly combined with numeral determiners to specify an amount) and definite noun phrases to refer to the ingredients in the development or in further setting conditions.

In contrast, the grammar of formal mathematical language is much simpler. Here, the indefinite introduction of an element is signaled by some special word like “given” and the reference problem is solved by assigning a unique name to the ingredient. An adaption of the original recipe to this form would be, for example:

Given \(p\). Assume \(p\) is a sheet of paper and \(p\) is quadratic. Begin with folding \(p\) in half…

However, in formulations which mix natural language and named references, it would be more common to simply add the name to the indefinite noun phrase as in this example:

Let a quadratic sheet \(p\) of paper be given. Begin with folding \(p\) in half …

Using short names like \(p\) as references for indefinite elements is particularly useful if several ingredients of similar type are used. While mathematical formulations stay very concise, natural language recipes involve rather lengthy noun phrases like “the first sheet”, “the second sheet” and “the third sheet” instead of \(p,q,r\).

Another advantage of named indefinite objects is the straight forward description of the application process. To check whether a collection of object descriptions fits to the setting of a mathematical recipe, the ingredient’s names are set in one-to-one correspondence with the object expressions. The additional setting conditions are then rewritten by replacing the names with the associated expressions. Afterwards, the resulting statements are checked. If they are satisfied, the object constellation under consideration is in the recipe’s range of applicability. Now the actions in the development can also be rewritten with the object expressions in place of the corresponding names. Carrying out these actions leads to the specified result with the same substitution of names by expressions.

In summary, application of a mathematical recipe turns into a symbolic manipulation of the recipe’s text: ingredient names are replaced by object expressions.

Since recipes with a reasonable range of applicability can be used with varying object constellations, the ingredients are replaced by varying expressions in these applications. For that reason, the ingredients in mathematical recipes are frequently called variables. However, this name is somewhat unfortunate as it either suggests some mysterious variability property of named ingredients or at least stresses the fact that we don’t know all properties of these objects which may appear uncomforting.

Due to these misleading connotations, we avoid the name “variable” which is also uncommon in everyday recipes. Who would call the tomato in a cooking recipe a variable? Everyone is aware that “a tomato” in the ingredient list has to be replaced by some actual tomato during application and by some other tomato in the next application. And everyone knows that ingredients are of course indefinite because we want to use the recipe in varying situations! There is no need at all to stress this obvious variability.

Predicates and sets

The setting of a recipe can be satisfied by certain object constellations. In this sense, a setting gives rise to a predicate which labels objects with the “fits” property. Similarly, we can think of a setting’s range of applicability as a set.

Conversely, any set or predicate can be described by a setting. In particular, evaluating predicates or determining membership to a set requires the same pattern recognition that is needed for recipe applications.

As a simple example, we consider the setting which admits all real numbers in the unit interval:

Given \(x\in\mathbb R\). Assume \(0\leq x \wedge x\leq1\).

Here, the name \(x\) of the ingredient is introduced together with the first assumption \(x\in\mathbb R\) which reminds of the typical noun phrase used in natural language descriptions like: “Given a real number between 0 and 1.”

The set representing the range of applicabilty of this setting can be denoted as \(\{x\in\mathbb R:0\leq x \wedge x\leq1\}\). Obviously it is built from the same syntactical components as the corresponding setting.

In mathematics, it is quite common to work with fictitious recipes. In fact, solving an equation often starts with assuming a solution was given. In the development, we then try to gather information about the solution components and, if possible, capture them in a meaningful representation. As an example, consider the following solution setting:

Let \(x,y\in\mathbb R\) satisfy \(2x + 3y = 5\) and \(x-y = 1\).

A development of this setting inevitably leads to the state \(x = \frac{3}{5}\) and \(y = \frac{8}{5}\). In other words, if we had admissible ingredients, than the recipe shows that they have to be \(\frac{3}{5}\) and \(\frac{8}{5}\) (this is a uniqueness result which indicates that at most one object constellation is admissible). To see that the two numbers are actually admissible, we just have to check whether they satisfy the two equations (which is an existence result).

Functions

A mathematical recipe whose result is an object can be viewed as a function: any object constellation from the range of applicability (the domain of definition) gives rise to an object which inevitably follows from the actions encoded in the development.

In the case of object formation, the associated actions are definitions which may be
accompanied by proofs that the corresponding objects are well-defined. The result is specified by highlighting one of the defined object names.

As an example, consider the function \((2x+1)_{x\in\mathbb R}\). It assigns to any real number \(x\) the value \(2x+1\). As a recipe, it would be described by:

setting
\(\quad\) Given \(x \in\mathbb R\).
development
\(\quad\) Define \(y:=2x+1\).
result y

Theorems, theories, structures and models

A mathematical recipe whose result is a proven statement can be considered as a theorem. The ingredient setting plays the role of the theorem’s assumption and the actions in the development are mainly proof steps. One of the proven facts is singled out as the result which is traditionally called the conclusion of the theorem.

If a setting leads to many different results and objects in its development we would call the construct a theory. Compared to a theorem, the result of a theory is not a single object or statement but a dynamically growing list. Structures like group, field or vector space are examples of theories.

Within the setting of a mathematical model, objects in the real world are described as abstract ingredients that are assumed to have certain properties. During development, consequences are derived that lead to further objects and statements which can answer the questions that are part of the modeled problem. The result of the model could be a composite statement consisting of all answers found.

Summary

Doing mathematics requires a foundation, which is basically a set of actions that produce new objects from given ones (term formation) and a set of actions which generate new facts from available facts (formula formation and proof steps).

To start the game, we just need some additional ingredients: inital objects and initial facts (axioms). Since the existence of these objects and the truth of the axioms are assumed, we can consider the beginning of mathematics as the setting of a theory which is subsequently developed with the available actions. By forming recipes about the ingredients of the initial setting, an increasing wealth of experience can be formulated and conserved. Moreover, the currently most widely used foundations of mathematics are based on objects like sets and functions which are inextricably connected to recipes themselves.

In other words: mathematics is all about recipes. Particularly, the basic recipe actions detection, formulation, matching and application are executed with the utmost precision!

However, when it comes to application of mathematical recipes to real world objects, the precision is lowered. In fact, being completely defined by a few assumed properties, mathematical objects are fictitious. As a consequence, matching to real world objects always involves some error which is typical for real world recipes as discussed above.

Nevertheless, mathematical recipes offer a major advantage: Since all basic rules and initial settings are transparently specified, and the validity of all actions can be systematically verified, their reproducibility is 100% guaranteed. Therefore, if the application of a mathematical recipe to a real-world object constellation deviates significantly from the promised result, the problem lies in insufficient matching rather than an error in the recipe formulation. In particular, it requires a new modeling cycle, resulting in a recipe that better matches the intended object constellation.