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Written by Dr Yes
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How to Prove Anything With Mathematics
1. There's No Such Thing as a Free Lunch
Dr Yes,
I've often heard the expression "There's no such thing as a free lunch". I like it, it feels good to say. But is it true? Is there any way to prove it? Perhaps you can prove or disprove it with mathematics.
Regards
Mandroid3000
Dr Yes answers: There is no such thing as a free lunch: an informal proof by someone who didn't go to their math lectures.
Or, How to prove anything with mathematics. Part 1.
The € can be read as a dollar sign if that pleases you.
There are two things to consider in the statement "There is no such thing as a free lunch". What is free? And, what is lunch?
I think we can define lunch as a meal eaten during the middle of the day. If you disagree then there is little hope for either of us. More difficult to define is free. dictionary.com (the dictionary of mathematicians?) defines free as follows (this is but a selection of the many definitions offered):
1. Not imprisoned or enslaved. Not subject to external restraint. Unobstructed.
2. Costing nothing.
3. Given, made, or done of one's own accord. Not controlled by obligation or the will of another.
4. Empty, unoccupied.
One could argue that lunch, being comprised of matter, is subject to external restraint. Namely the lunch is restrained by the laws of physics. if you drop your lunch it will fall to the floor, thus it is restrained in this case by gravity. Therefore, if we use the first definition of free, we can say that lunch is not truly free. We will dismiss this point however as it is more than a little obtuse and not in the spirit of fair play.
We have probably all had a meal, during the middle of the day, which we have not paid for in a material sense. Was it a free lunch? It has cost us no money and thus was free in terms of definiton 2 above. This could be taken to be a free lunch but has the lunch truly come with no obligation towards whoever provided you with the lunch? If the lunch was paid for by a friend, you paid no money for it so it was free in a monetary sense, but you may be obliged to pay for lunch the next time you meet. We can consider this a defered financial cost. Additionally you were probably obliged to listen to their pointless stories during the duration of the lunch and perhaps for some time thereafter. Thus, while lunch may come with no financial cost, either immediate or defered, it almost certainly has strings attached and thus is not free in terms of the obligations you feel towards the provider of said lunch.
Where does the mathematics fit into this rambling? Well we can define the cost of lunch as:
cost (C) = financial cost (€) + non-monetary obligations (o).
This assumes a scientific definition of free (definition 4 above) whereby free can be thought of as unoccupied or zero. Thus using this definition, if C = 0 then lunch is free, if C ≠ 0 then lunch is not free.
There are two cases where the lunch can be free. These are: (a) when there is neither a financial cost nor any obligations felt to the provider ofthe lunch (€ = 0, and o = 0); or (b) when the sum of the financial cost and the magnitude of ones percieved obligations are 0 (€ + o = 0).
Both cases can lead to a free lunch. A completely morally deficient person who feels no obligations towards the provider of the lunch (o = 0) may have a free lunch if it is paid for by somebody else (€ = 0). If the eater of the lunch percieves any obligations toward the lunch's provider (o ≠ 0) there is hope for them too. They have the possibility of a free lunch if the financial cost of the meal is negative. i.e. if they are paid to eat the meal. This will result in a free lunch if -€ = o. Note that if -€ > 0 then cost < 0 and lunch is not free either.
This is good and well in the realm of mathematics but is it really possible to have a free lunch? As it is probably impossible to put an accurate monetary value on the obligations one feels towards the provider of lunch (estimating -€ = o) it is highly unlikely that case (b) will result in a free meal as one will inevitably receive too little or too much financial remuneration, whereby -€ ≠ o and C ≠ 0. Thus, for practical reasons case (b) can be ruled out.
Case (a) requires somebody buying a completely amoral person lunch. I also find this possibility highly unlikely and thus must conclude that there is no such thing as a free lunch.
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