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tags: mth350
---

# A word about "backwards" proofs

:::warning
**TL;DR:** "Backwards" proofs --- ones that start with the conclusion and "work backwards" until it reaches a known fact --- up to this point have been considered to meet the homework objectives as long as everything else was OK. **Starting with Homework 6, and all revisions from week 8 onwards, this will no longer be the case, and "backwards" proofs will need to be revised so they are "forwards".**

:::

## Backwards proofs and why they are bad

A "backwards" proof of a theorem is one that starts with the conclusion of the theorem and works backwards until a known fact is reached. Then, the logic states that since we have reached a known fact, the backwards chain of reasoning is activated and therefore the conclusion is justified.

Here is an example of a backwards proof, using Theorem 1.2.3 from your textbook.

:::info
**Theorem 1.2.2: Let $a,b,c \in \mathbb{Z}$. If $a|b$ and $a|c$, then $a|(b+c)$.**

**Backwards proof:** Start with $a | (b+c)$ and work backwards. If $a | (b+c)$ then by definition there is an integer $x$ such that $b+c = ax$. We also know that since $a|b$ and $a|c$ that there exist integers $q,q'$ such that $b = aq$ and $c = aq'$. Adding these together gives $b+c = a(q+q')$, and since $q+q' \in \mathbb{Z}$, we know that $b+c = a(x)$ for some $x \in \mathbb{Z}$. Therefore the theorem is true. $\square$
:::

There are two things that are bad about backwards proofs.

### 1. They are not as readable as "forwards" proofs

You might find my backwards proof above to be confusing and weirdly organized. I would agree. Compare it to the "forwards" version:

:::success
**Theorem 1.2.2: Let $a,b,c \in \mathbb{Z}$. If $a|b$ and $a|c$, then $a|(b+c)$.**

**Forwards proof:** Assume that $a|b$ and $a|c$. Then there exist integers $q,q'$ such that $b = aq$ and $c = aq'$. Therefore $b+c = aq + aq' = a(q+q')$. Since $q+q'$ is an integer by closure, we have that $a | (b+c)$. $\square$
:::

This is much shorter, simpler, and direct. So realize that

:::danger
Every backwards proof can be rewritten into a fowards proof that is easier to read.
:::

### 2. Backwards proofs are often logically invalid

Backwards proofs start with the conclusion, which is never a good idea because this can lead to absurd statements even though each step can be justified.

For example, here is a "proof" that $0 = 1$.

:::info
**Claim: $0=1$.**

**Backwards proof:** Starting from $0=1$, multiply both sides by $0$. We get $0 = 0$ as a result. Since this is clearly true, we must have $0=1$.
:::

Obviously $0$ does not equal $1$, so what was the problem? **Starting with the conclusion, was the problem** because the conclusion itself is false. So realize that:

:::danger
Backwards proofs always contain an implicit assumption that the theorem you are trying to prove is already true and therefore constitute invalid reasoning.
:::

There is a way to write a backwards proof that is logically valid, namely if each step in the proof is true "if and only if" the preceding and following step is true. But this is difficult to do, and readability is improved by writing it "forwards" anyway.

## No more backwards proofs from here on out

Up through Homework 5, "backwards" proofs were sometimes allowed to meet the [Homework Objectives](https://docs.google.com/document/d/1VQmw_F5nJZGqAJmhTdgEBDIy9ZY2qTWtqe3QdZ9Km_Q/edit), but to make an otherwise correct proof "exemplary", you were asked to rewrite the proof "forwards". **Starting with Homework 6 and subseqent homeworks, and with all homework revisions, backwards proofs are no longer accepted as meeting the objectives.** Such proofs will need revision in almost all cases.

**Using backwards steps in an outline is OK:** You are taught in MTH 210, and we emphasize it in MTH 350 as well, that it's often a good idea when outlining a proof to write down the assumptions and then write dquareown what you want to prove, and then make **forwards steps** from the assumptions and then perhaps make **backwards steps** from the conclusion until the entire argument begins to take shape. **This is a good idea --- for the *outline* of a proof.** But when writing a finished argument, it needs to flow forwards.